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Y2, Y1. -+- 1 2
X2 *7/ ) (/-: ..26/28(1 ) (/-: (3 9 +-1. ,2) -7 3,': )135 6. (+-+13 -.) -5 8(113 , (3 -50 -13 ) (/-:3 -7 ')2 3-3 ');,
*--(1 195 -23.3 ' ., *-7')1 (:, 35 (:-3 .--5 '. (- 3)4 32,)2 -5 3((31 (9 3-5 ('.;6(:'793 '), 35-43 .--5( .'()4.3 .--5 ('.;2 -5 3((31 397 (') 6'55 . (+-+13 27 -7' 1 26. (-',:-2 . (7'1 '+ *'51( *:(-;, *--,'4, *-7-23.5 4 ( : '(43 2) -'4 -3( -) -2)3 42/5 *-7' -25 * -7-23. 5(4'1 . (')') ,) (:5 5-/': 6. (:(7. ( * -0 :1 (1 *- (2-; +-+5 195 * -7-23.5 2-/.:* -7-23. 2 . (-',:-2 . (7'1 .;)3 22(7 -8' 195 *-7-23.5 4 (: (52 6+/, +8 2) ('79( * --0'4+6(2,7
.(:(7. ( * -'51 ) ' (-;, 3+(-+3 ) ' +:3 )' - -53 -24-9-;3 ) '3 (-;,5 4 ( : 39 42/5 6* -'5 1 .')') 2) ) '3
.('5.3 5/'15/'1 3-5(4 2) 3'415 31 (+ 6-( -: 2) . (-');,3 . (,8(.3 27 ., 22(73 (,
Ω = ω * +13 5/'1 ) ' 2) 3'415 63+-/-3 21 4-3 2 .(+(4 :3 (, 302(' 2) 3'415 6* -0 :12, * 5/'1 , (3 -');, * +15/'1 6[0, 1]
193 045 . (;-8'3 . (-84 :(;3 27 (, . (-32 2(7 - * +1 5/'1 [0, 1]
193 045 '511 3,-8-3'(/52 * -2 (7 - (:, '), 2+(1 (39 227 '+5 , .-');, 3,8(. 27 ',.2 -+7 4 -;1 '-) .(-32 5--/ * +136) ('+3 * (1 -:-131 2(+ 3-3-) 7 (. (, '(/52 / (: * -. -2(2542 .-:) +-13 ., ',.1 . ('(,13 5/'1 6
Ω2) . (8(54 .. 2) (,
F = A .('(,13 5/'1 (:, 3 -5(4 2) 3'413 '(5 2-2 . (,1 (+5 6* -(1 * +1ω1
'1 (27 ω1 ∈ Ω
3-3- &*--( -:3 . (,8(. . (+(,72( ;, ,-3 *-(1
ω1.;(32 . ('5.33 . ('/,3 . (,1 (+3 -.)5 6;,1 3:() . ('5.3
ω12 /-2 * -2 (7 -
) .('5.33 31 '5131 )'3 '(5 2(,)2 27 (: ,1 (+2 6* -1 +2 ,2( . ('(,12 /-2 (:-2 .('5.33 .,Ω
2) . (8(543 .. (,) ) ('+2 ) - . ('5.3 2) . (51 3'(. . (:52 3-3 - ');,) .:1 2 6∫ 1
0 n2(t)dt < 3
*-,53 *-,:.3 ., *--4 -F = A
6'(,1 ,-3Ω2)
Ω38(543 .. '1(27
Ω ∈ F 66(A1 ∪A2 ∈ F )
'(,1A1 ∪A2
* -9, Ai ∈ F 6, 69 . ('(,1
A1, A2*, 6
6F
2 --)Ac
i* -9,
F2 --)
Ai*, 6
6∪Ai ∈ F * i = 1, 2, · · ·Ai ∈ F *, '1 (27 '(,1 , (3 , .('(,1 2) . (:132 .-: +(/, 6
6 (. -/ ./. '(F
-7 5(: (2, . () -'+1* -:.:3 * -2+ 27 . ('(,1 2 5()/2 ');, 6
Ω2) . (8(543 .. 27 ., 2-732 5--/ (:-,
F (,3
1- -25 . (. (, -:) -5 (:-/5- ,2) 7 . ('(,13 ., '(/52 .-: -9, 3-':, *-++(1 (:, 2)12 *, 7 63+-+12. ('(,1 5/'1 , (3F
.=
Ω, ∅, 1, 2, 3, 4, 5, 6 (,3 (
Ω = 1, 2, 3, 4, 5, 6 3-5(43 2) 31 (+5 2)12 6 (;364(+5 -1 -0 -2. ('(,1 -7 52 *-) 6* -'9 . ('(,1 *-,'4:
A ∩ B = ∅ *-1--413 (A ∈ F,B ∈ F )A,B.('(,1
(+- -9,A3'4 *, 7) )135 3'+3 3,' .-0 -00 *--(2. +-1. *3 .-5 (-/ . ('5.3 *3-:)2 '), *-'96
B3'4 ,2 -7 . (,+((5
P (Ac) + P (A) = 10 ≤ P (A) ≤ 1
*--41P (A)
A ∈ F '),7
A'(,1 27 '(5
P (A).('5.3 .-84 :(;6
P (Ω) = 17 (17
P (∪Ai) =∑
i P (Ai)-9,
i 6= j27 '(5
Ai ∩Aj = ∅ A1, A2, · · · (. : *,6. ('5.3 5/'1 .,'4: Ω, F, P .(8(54 '(5 .'+ (1 /'735 ,2(
F2) *-0 :12, '1(27 . ('(,1 '(5 4' .'+(1 . ('5.33 .-84:(; '(1,76.'+(1 3 -3. ++(5 * + ω 2) . ('5.33) /'73 -, 0';5 6. ('/,
, (3 ω : X(ω) ≤ a -)11 a27 '(5) 7 * -1 +3 5/'1 2
ω ∈ Ω, X(ω)3-84:(; , (3 , 1 -,'4, 3:.)16. ('(,1 +-1. 254: , 13 -7' -+- 2 -( -: . (,8(. +(+1: *, '1(27 6'(,1
3;-8' 2 .'(8 , (3n(t)
'5131 3,-8-3 )') /-:: '513 )' 2 , 1 2) . (,1 (+6t = 0.25
'5 )'3 n(0.25) = n(t)|t=0.25
66([0, 1]
193 * (/.5 ) '3 . (, 2) 3-':,3 X =
∫ 1
0 n2(t)dt
66[0, 1]
193 * (/.5 ) '3 . (, 2) -21 -713 '3 Y = maxt∈[0,1] n(t)
66) '3 2) *-1 +3 . (;-8' 2 .5.1 3/7 (33 3/7 (3 . (:(0 3 , .(-5-0 -, (0 :-, 3 . (,1 (+5 . (:03
2) 3-84 :(;7 (9 . ('5.32 6. ('5.3 (' (5 '-+32 .-: '(,1 , (3 ω : X(ω) ≤ a , 1 .'+3 -;2) ((-712) . ('5.33 4 (/ ., 3'-+1X(ω)
-,'4,3 3:.)13 2) (2-;3 .-84 :(; 6X
2) (2-;3 . -84 :(; *-,'(4a
X(ω)
FX(a) = ProbX(ω) ≤ a6-1 -1 3;-8' ,-3) /-7 (32 .-:( .+'(- ,2 .-:(0 (:(1 ,-3 4+:-, 4' ,(3X
a2) 3-84 :(;7 +-1. .'+(132(
FX(a)+-+5 , 1 '(5 6
FX(−∞) = 0, FX(+∞) = 1*-,53 *-7'3 ., (2-;3 .-84:(; .2541 . (2(55
fX(α)-9,
α272
FX(α) =∫ α
−∞fX(θ)dθ
) 7fX(α)
*--4 *, 64/)1 .-5(4 '(5FX(α)
., '--8 .(8-;456FX(a)
2) .'9:3 ,-3fx(a)
-9, '-9FX(a)
*, 0';5 6X
2) ; -2 (3 (2-;3 (, . (;-;83 . -84 :(; .,'4:'(,13 -7
Proba1 < X ≤ a2 = FX(a2)−FX(a1)*--4.1
a1 < a2'(5) +-1 5(: (2-;3 . -84 :(; .'+316
FX(a)2) . (-:(0 (:(13 .5 (: * ,71 6X ≤ a1 ( a1 < X ≤ a2 *-'93 . ('(,13 2) +(/-, ,(3 X ≤ a2
αi
mx = X = E[X ] =∑
i
αiProbX = αi ,
6∑i |αi|ProbX = αi <∞ ) -,:.5 '1 (27 50 -3 '+(1 -(0 -53) -,:.5(
mX = X = E[X ] =
∫ ∞
−∞
αfX(α)dα
5 -(2.m
'),7 i = 1, 2, · · · ,m ; α
(n)i
3-3. n27 '(5 -227 ;(,5 6∫∞
−∞ |α|fX(α)dα <∞ ) 3/:35 .,9(6, 69 n→∞ '),7
maxi |α(n)i+1 − α
(n)i |−→0
*--4.1) /-:: 6α
(n)i < α
(n)i+1 ; (−n, n)
043 2) .-;( 34 (2/ n6'. (- 3:-+ .-) : 34 (2/3 2+
n) 277'-+:
E[X ] =
∫ ∞
−∞
αdFX(α) = limn→∞
∑
i
α(n)i
[
FX(α(n)i+1)− FX(α
(n)i )]
3:('/,3 3'+33 ; +-+5 *-1+(43 *-'413 '(5) 4 (+5 6. (4 (2/3 .'+5 -(2. (:-, ( * --4 2(53) 3/:3563:(7 :3 3,8(.3 ., .:. (: 7,
6 '(-,6E[Y ] = E[g(X)] =
∫∞
−∞ αdFY (α)3'+33 -;2 -9,
Y = g(X)3-3 -
3/7 (3 ,22 35()/ 3:0.-84:(; ., *+(4 5)/2 '(8 -, Y2) .2/ (.3 ., 5)/2 .:1 2 39 3'415 (:--3+
E[Y ] =∫∞
−∞g(α)dFX(α)
'+3 X
2) (2-;3 .-84:(; . -+- (.1 . ('-) -EY
., 5)/2 ');, 6EY
., 5)/2 39 (2-; (.1 (Y
2) (2;363'84 '. (- +-1. 017 ,-3 3-:)3 ; * , 1 '(5 ,1 (+2 6.2/ (. ) - , 1 272 ,2
fX(α) =
0, α ≥ 02/π
1 + α2, α < 0
.
6∫∞
−∞αfX(α)dα = −∞ *--4.1
) -'. (- , -(2 (.; 3'41 7.- 6. -; ( .2/(. (2 -,) ,-3 3:((73 (.2/(. -, , 12 -7 '1,: 39 3'415,-3 . (;-;83 .--84 :(; *, 2)12 6*--4 ,2 2-2 2(53 '),7 2'0:-,3 ., '-+32 227 .-: ,2) *-'41fX(α) =
1
π
1
1 + α2
6
, (3 * --2-2) * -7' '(5 2'0:-,3 (2-, ( ∞ , (3 * --5 (-/ * -7' '(5 2'0:-,3 7) .'+(1 3:-, .2/ (.3 -9,6(−∞)
142+7 +525 (, *-7' 25413 , 1IA
5 1 :( '(,1A
3-3- 6,IA =
1 ω ∈ A0 ω ∈ Ac
./ (:1 . ('-) - (, .2/(.3 .'+31 (A
'(,13 2) '(04 -+:-,3 . -84 :(; (, .:--813 3-84 :(;3 ,'4:IA254: +-+5 , 12 .2/(.3
EIA =
∫
αdF (α) = 0 · Prob(Ac) + 1 · Prob(A) = Prob(A)
*--4.1 -9, ((:1 -,'4, 3:.)1 . ('/, *-2-15 (, -0 -:-1'0+X
'1 (27 X = const
*, 656E[constant] = constant6(,2 *, *--(2.
X,Y*, 5()/ ,2(
E[aX + bY ] = aE[X ] + bE[Y ]-9, 6* - (54
b, a, 1
X,Y. (-',:-2 6
6EX ≥ EY -9,
ω272
X(ω) ≥ Y (ω)*, . (-:(0 (:(1 6+
E[X2]-:) 0 :1 (1
E[Xn]n'+1 0:1 (1
E[|X |n]n'+1 02/ (1 0 :1 (1
E[
X −X]
= 00';5 (
E[(X −X)n]n'+1 -97'1 0 :1 (1
E[(X −X)2] = Var(X) :,-' ( .(:()
σX =√
Var(X)4.3 .-0
621'(:1 , 1Z
-9,Z = X−E[X]
σX
*, 72(E[X2] = 1, E[X ] = 0
*, 21'(:1 , 1X
39 -,'4, 3:.)1 2 +-32 ');,) 31 27 , (3 (FX(a), a ∈ R)
.('5.33 4 (/ ++(5 -,'4, 3:.)1 '(5(:-,E1/n|X |n *,(
EX,EX2, EX3, · · · *-:(. : *, (;33 ((75 6EX,EX2 · · · 2 '5+2 ');, .,9 . (5456
X2) . ('5.33 4 (/ ., *-'-+1 *-0 :1 (13) . (,'32 ');, -9, -+1 '31 32(
*-0 :1 (13 -:) EX2 .,(
EX., .+2 4 -;1 --:1 ,2( (+- ,2 . ('5.33 4 (/ . (-:70 . (-5 35'3 '(826. (-53 ('.; ');,2( . -227 3:(1. ..2 *-2(7 -3 .('5.33 4 (/ ., *-'-+1 *:-, 227 '+5) *-:(),'363,-)3 2 35(0 37'3 . (:
E(X − Y )2/EX2 -9, 6X
2) 3+-+1 ',.13 , 1Y
, 1X
2)12
5 8-5 8 2) (-(()3 -,2-2 +( , .(:(7. -;2
E[X2] ≥ E[
X2 |X|≥ε
]
≥ ε2E |X|≥ε
= ε2P (|X | ≥ ε)
,71 (P (|X | ≥ ε) ≤ EX2
ε2.
,71 (EX2 = VarY
-9,X = Y − EY /4: *,
P (|Y − EY | ≥ ε) ≤ VarY
ε2
.-: 7 6. (-('5.3 5)/2 '),1 , 1 2) -:) 0 :1 (1 5)/2 '. (- 24 *-5' * -'415) ((-7 -) (1 -) 39 (-(() -,. (4 -'9 (.1) . ('5.33 5() -/ 31 (+2 68(1131 3 -'/ 2) . ('5.3 2 (-2 */ 37'3 25425)/2 24 , 3)4 ,(3 8(1131 2) 3-0 '1 (27 ./.1 (, 21 3-3- & 3 ';1 50168.24
0;)1 3,' '. (- -227 0;)1 2) -0'; 3'41 (39 65 8-5 8 */, 1 2) .-:-;(,3 3-84:(;3
- .'+(1 ΦX(ν), X
, 1 2) .-:-;(,3 3-84:(;3ΦX(ν) = E[eiνX ] = E[cos νX + j sin νX ]-9,
fX(α)-2 ( (2-; ) -
X-,'4,3 3:.)12 *, 6
j =√−1
'),7 ΦX(ν) =
∫ ∞
−∞
ejναfX(α)dα
+/ 3'-+1φX(·) .-:-; (,3 3-84 :(;3 3'41 275) . (,'32 ');, 6
fX(α)2) 3-' (; .'1.3 ,-3
φX(−ν) (:--3+6FX(·) (2-;3 .-84:(; ., .-1)1
(:-3 . (1-/ *--4 +-1.E[| sin νX |]( E[| cos νX |] -7 * --4 +-1.
ΦX(ν)*2(, *--4 ,2
E[Xn]) 7. -* *(7-3 '+ ., -2/32( . (4 9/ '(02
ejνX ., 4';2 27 (: *-0 :1 (1 * (-4 . (1-,.1 . (/:35 6 (:- (43 (254 :( .2/ (.3ΦX(ν) =
∞∑
k=0
(jν)k
k!E[Xk] .
(:43 6(2-;3 ., (542 .-: 37 (.1 '(1,7 ( .-:--;,3 3-84:(;3 ., 5)/2 ');, *-0 :1 (13 .'+ (.1 '1 (276(2-;3 ., 5(4 * -0 :1 (13 (, *-1-,.13 *-,:.5 -7
254:( ν-;2
ΦX(ν)'(5 ('/,3 -(0 -53 ., '(9: (7 ( * (-4 -,:. 4 (+52 -25 5() (
φX(0) = 1;∂φX(ν)
∂ν
∣
∣
∣
∣
∣
ν=0
= j EX ;
(
∂nφX(ν)
∂νn
)
ν=0
= jnEXn
φY (ν) = ejνbφX(aν)-9,
Y = aX + b*, 3:0
3/7 (3φY (ν) = E[ejν(aX+b)] = ejνbE[eiνaX ] = ejνbφX(νa) *--,'4, *-:.)1 -:)
, 1Y,X
*,FX,Y (a, b) = PX ≤ a, Y ≤ b
fX,Y (a, b) =∂2F
∂a∂b
E[g(X,Y )] =
∫∫ ∞
−∞
g(α, β)fX,Y (α, β)dαdβ
6Y
*X
2) *-52()13 *-0 :1 (13 ., 254:g(X,Y ) = XmY n '(5 +/ (-15 (
*--4.1 *--)11a, b
27 '(5 *, .-0 -00 . 5 *--(2. -.25 *-,'4:Y,X
, 1 -:)PX ≤ a, Y ≤ b = PX ≤ a · PY ≤ b (,
FX,Y (a, b) = FX(a)FY (b) .
2 2(4) 39 (-(() -9, . (;-;8 ) - * -:.)12 *,fX,Y (a, b) = fX(a) · fY (b).
5(: ,71 6Eg(X)h(Y ) = Eg(X)Eh(Y )
*--4.1g,h
.(1 (/ . (-84 :(; -.) 272 * 1, . 5Y,X
3/7 (3 ,22* --4.1Z = X + Y
( * --(2. -.25Y,X
'(5φZ(ν) = φX(ν) · φY (ν)
- '+(1Y,X
, 1 (9 2) :,-' ((43Cov(X,Y ) = E(X −X)(Y − Y )62 .5 .-',:-2 * --(2. -.25 (, 3-82'(4 -'/
X(Y
, 13) *-'1 (,Cov(X,Y ) = 0
*,/'735 (:-, ;-33 25, *-1--4 * -0 :1 (13) 3/:35 .-',:-2 . (2. -, .5--/1 . -0 -00 . (2. -,) /7 (3 2-'.6(7 :
X,Y
X.= sin Θ, Y
.= cosΘ [0, 2π]
Θ
1.2
3-82'(4 *+41E(X1 − λX2)
2 -(0 -53λ39-, '(5 6
EX1 = EX2 = 0.(0);3 12 /-::
X1(X2
*--,'4, *-:.)1 -:) '(5 -21-:-1 3-3- . - (5-'3 3,-)3 8(11) 7
X22) .-',:-2 3-84:(; .'95
X1., 5'42 .-: +8-7 (, -21-:-13-89-1-:-13 ., (:-) (-2 -(0 -52
λ∗., -8: *, 6-21 -0;(,3 , (3
λ∗ = Cov(X1,X2)Var(X2)
) 254:λ-;2 3'-9 -254:
E(X1 − λ∗X2)2 = EX2
1 − 2λ∗ Cov(X1, X2) + (λ∗)2EX22 = VarX1 −
(Cov(X1, X2))2
Var(X2)≥ 0
&'(() -) (4 (-(() -, ., (:254 ,71(Var(X1)Var(X2) ≥ (Cov(X1, X2))
2
*+412ρ =
Cov(X1, X2)√
Var(X1)Var(X2)*, 6ρ = 0
-9, .-',:-2 * --(2. -.25X1, X2
*, 63:('/,3 3,8(.3 '(,2 |ρ| ≤ 1( 3 -82'(43 *+41 *-,'(4.-',:-2 . (2. ) - (2, *-'415 6-2-2) ,(3
α-9,
ρ = −1*,( (3)27
α-5 (-/ *+41 '(5
X2 = α ·X1-9,
ρ = 1635 4 ( :) . (-5 ('.;5 -;(1 39 -(0 -5) 3,': )135 6-'52,3 5(15
, (3bT a =
∑
aibi-9, 6* --+1-1
n*-'(04 (
a, b(-3 - 6. (8-'01 (:1 -
B ,A63 +(1 -' (04 ( (:1 -
X, a*-:(1 -'),7
A ·B 2-;732 ');,.1A ,B
.(8-'01 .2;71 -7 '79: 7) n× n 38-'01 ,-3
a · bT .,9 .1 (2 6'24
,1 (+2 6m× k ,-3 .254.13 38-'013 6
n× k 38-'01 ,-3B
( .(+(1n.('()
m m× n 38-'01 ,-3
A
a1 a12 a13
a21 a22 a23
b1 b12
b21 b22
b31 b32
=
a1b1 + a12b21 + a13b31 a1b12 + a12b22 + a13b32
a21b1 + a22b21 + a23b31 a21b12 + a22b22 + a23b32
*-5-7'n25 -,'4, '(04 (
X-3 -
X =
X1666Xn
; XT = (X1, X2, · · · , Xn)
6* --4 , (3 '),7fX
; .(;-;83 * (17 *-:.)1n2) 3-84 :(; -3 (9
FX1(. '(04 ((3 2) (2-;3 .-84 :(;-+- 2 '(04 (( 2) .2/ (. '-+: 62 -2 , 1 (9 ' (5 (17 ,-3 *.'+3
EX.=
EX1666EXn
.
, 1 *3 3-'5, '), 38-'01 2) .2/ (. 31(+ 3'(85 6. (2/ (.3 '04 (( . (-32 .'+ (1 ' (04 (( 2) .2/ (. '1 (276* -1-,.13 , 13 2) . (2/ (.3 *3 3-'5, '), 38-'013 . (-32 .'+(168(113 '(04 ( , (3
EX( (),' '+1 *-0 :1 (13 * -,'4:
i = 1 · · ·n,EXi
6(),' '+1 *-02/ (13 * -0 :1 (13 * -,'4:i = 1, 2, · · · , n, E|Xi|
E[
X ·XT]
= E [XiXj]ni,j=1
.-8-'01 3'(85 6-:) '+1 *-0 :1 (13 * -,'4:1 ≤ i ≤ n, 1 ≤ j ≤ n,EXiXj
6-:) '+1 *-0 :1 (13 .8-'01 ,-3'1 (27 -:) '+1 *--97'13 *-0 :1 (13 * -,'4:
1 ≤ i ≤ n, 1 ≤ j ≤ n'(5
E(Xi − EXi)(Xj − EXj)-8-'01 (1 -5 6* - :, -'((43
Cov [X,X].= E
[
(X − EX)(
XT − EXT)]
6E[
XT]
= [EX]T 3'+33 -;2 -7 52 * -)
X'(04 ( 2) . ('5.33 4 (/ 2(+
n'),7 *--,'4, *-'(04 ( ' (5
FX1,X2··· ,Xn(a1, a2, · · · , an) = PX1 ≤ a1, X2 ≤ a2, · · · , Xn ≤ an*-0 :1 (13 ., .+2 4 -;1 . (5' . (-52 *2(, 6 (+- ,2( '. (-5 57'(1 . (-32 2(7 -
FX(α)-' (04 ( 5-.75 (, '+1( (),' '+1 *-0 :1 (13 *-:(. : 3,53 3-53 ,-3 )135 35 4 ( :) . (-53 ./, 6-:) ( (),' '+1
142+7 '+(1Y
'(04 (3 (X
2) -:)Y = AX + b';12 3() . (-32 5(17 -'8
A2) . (+(13 ';1 6 (54 '(04 ((
b( .-,'4, ,2 3 (54 38-'01
A'),76
b = 0-7 . (0); *)2 /-:: 6
X2) *-0 :12, .('()3
32,)2 31+43 5 (17 ,-3 (9 32,) Y
2) -:) '+1( (),' '+1 *-0 :1 (13 ., , (812 ');, (. :3 (.1 *,3 (.1 5)/2 .-: *,3 ,-3 ,7 32,)3 6. -',:-2 .7'1 ,-3
A( -,'4, -23. , (3
X'),7 '. (- . :--:163 -:75 -:)3 ( (),'3 0:1 (13 2 +-15 4' ) (1 -) (.
(Y ),8(15 -:)3 0:1 (13 ( 8(113 ., *-:(. :3
. (2/ (.6EY = E[AX + b] = AEX + b
-9,Y = AX + b
*, 6* -1 -,.1 *-+1-1 -25b'(04 (
A38-'01 *-:(. : 3:0
6 '(-,6. -',:-2 32(; ,-3 .2/ (. '1 (27
,53 '-) -3 (5)/31 .5 (: 3,8(.3 3/7 (3
E
a1 a12 · · · a1n
a21666 666am1 · · · amn
X1666Xn
+
b1666bn
= E
a1X1 + a12X2+ · · · +a1nXn + b1666am1X1 + am2X2+ · · · +amnXn + bm
=
a1EX1 + a12EX2+ · · · +a1nEXn + b1666am1EX1 + am2EX2+ · · · +amnEXn + bm
= A
EX1666EXn
+
b1666bn
*--,'4, *-0 :12, * .(+(1n
.('()m
.-,'4, 38-'01Y
3-3. 3:('/,3 3:03 2) 35/'3
Y =
Y1 Y12 · · · Y1n
Y21666 666Ym1 · · · Ymn
= [Yij ]
.(,'2 24 -9, 6* -1 -,.1 *-+115 . (-,'4, ,2 .( (54 . (8-'01B
(A
3:-3. 6EY = [EYij ]
-7 '79:-52 .2/(. -5 -2/32 +-1. .-: '1 (27 6
EAY B = AEY ·B 72(EY B = (EY ) ·B 7 (
EAY = AEY)67 -.1 .-,'4, 38-'015 2;7 '(5 (7 : +-1. (:-, 5(17 '5+3 6. -0 -:-1'0+ 38-'015 2;7
-:) '+1 *-0 :1 (1E[
X ·XT]5 -- : 6+525 * --,'4, *-'(04 (5 4 ( : ,7( -:) '+1 *-0 :1 (12 .7 '(5:
E[
X ·XT ]]
= E
X1
X2666Xn
(X1, X2, · · · , Xn)
=
EX1X1 EX1X2 · · · EX1Xn
EX2X1 EX2X2 · · · EX2Xn666 666EXnX1 · · · EXnXn
'-+: 31 (+ 3'(85 6-:) '+1 *-0 :1 (13 .8-'01 .,'4: (9 38-'01 .-'01-n × n 38-'01 .254.1 52 * -)
:, -' ((43 .8-'01 .,
E(X − EX)(X − EX)T
=
Cov(Xi, Xj)
i,j=
E(Xi −Xi)(Xj −Xj)
det(EX ·XT ) = 0) (7 : /'735 ,2 39 25, 6 3 '+1 38-'01 (9) ((-7 ;, +-1.
det(X ·XT ), n > 1'(5 3'3'(5
EX2i = 1 EX1X2 = 0, EX1 = EX2 = 0
*, ,1 (+2 .-',:-2 32(; 3:-, 30::-1'0+3 .2(;) * ()1613(() 0 ::-1'0+3( 3+-/-3 .8-'01 ,-3 *-0 :1 (13 .8-'01 -9,
i = 1, 2,(:-,'3 '57 -9, Y = AX
- '+(1Y
3-3- 6E[X ·XT ]
7 (EX
(+-(X
, ( (. :) /-:: 3,53 3-55 .7 --:'1 (27
Y2) -:) '+1 *-0 :1 (13 .8-'01 -31 32,)3 .2,) : 6
EY = AEX) *+(43 -5
E[Y · Y T ] =?
,1 (+5 --:Y =
Y1
Y2
; A =
a1 a12
a21 a22
39 3'415
Y1
Y2
= AX =
a1X1 + a12X2
a21X1 + a22X2
. (: '-) - (5)/ (
Y1
Y2
Y1
Y2
T
=
(a1X1 + a12X2)2 (a1X1 + a12X2)(a21X1 + a22X2)
(a1X1 + a12X2)(a21X1 + a22X2) (a21X1 + a22X2)2
(5)/3 *2(,E[X ·XT ]
. -+- (.1E[Y · Y T ]
38-'013 ., 5)/2 ');,) '('5 3:('/,3 3,8(.5 . (27.3130 (); +(,1 3'(85 (5)/3 ., *72 (:2 ');,1 . (8-'01 2) 2;73 ( . (8-'013 ) (1 6-1 -+ 3,':
E[Y Y ]T = E[AX(AX)T ] = E[AXXTA T ] = AE[XXT ]A T
31(+ 3'(85 ((
Cov(Yi, Yj))
i,j= E
[
(Y − EY )(Y − EY )T]
= E[
(AX −AEX)(AX −AEX)T]
= AE[
(X − EX)(X − EX)TA T]
= AE[
(X − EX)(X − EX)T]
A T
= A[
Cov(Xi, Xj)]
i,jA T
6 '(-,*7 : -,'4, '(04 ( 2 .-',-:-2 3-81'(; :'0 * -851 '),7 -:) ( (),' '+1 *-0 :1 (13 2 .(,8(.3 .,6
1.9'(-85
-:) '+1 *-0 :1 (13 .8-'01 2) .- -5 3:(7.*--4.1
a-+1-1
n'(04 ( 27 '(5 *, .-2-2) -, .,'4:
(n× n)C
.-'01- .- (5 -' 38-'01aTC a ≥ 0.-5(-/ .,'4:
C-9, *-;,.1
a-5 -7' 27 '),7 4'( ,
aTC a = 0 (:5 *,( '24 , (3
aTC a) 52 * -) 6.'+(1 . -5(-/ (,
6. -2-2) -, +-1. -:) '+1 *-0 :1 (13 .8-'01 3:06. -2-2) -, +-1. ,-3 * ) 3,'- 0;)13 -:) '+1 *-0 :1 (1 .8-'01 ,-3 :, -' ((43 .8-'01) ((-71 3'3
72( -'24 -,'4, 3:.)1 , (3y-9, 6
y = aTX = XTa/4: 3/7 (3
0 ≤ Ey2 = E[aTXaTX ] = E[aTXXT a] = aTE[XXT ]a.(1)1 2 6. -'2(:- /'735 3:-,E[XXT ]
25, .-'2(:- 38-'01 +-1. 3:-3XXT (:'3 '57) -;7 3'3.-5(-/ ,-3 -9, .-'2(:- 3:-,
E[XXT ]*,) .(,'32 ');, 6)135 +(1: . -'2(:-
E[XXT ](5) 3'4136.'+(1
-,'4, '(04 ( 2) . -:-; (, 3-84:(;6+1 -1 (. (,5 -0 -:-1'0+ '(04 (
ν( -,'4, '(04 (
X-3 -
X =
X1666Xn
; ν =
ν1666νn
'-+:φX(ν) = E ej
Pnk=1 Xkνk = E ejXT ν = E ejνT X = E ej(X,ν).-84:(; 2) .-+11 5' 3-' (; .'1.3 -1 -3 (9 6* -' (04 (3 -:) 2) .-'243 32;713 ., 11
(X, ν)'),7-9,
fX.(;-;8 .-84 :(; ) -
X-,'4,3 '(04 ((2 *, 0';5 6.;. ()13 (2-;3
φX(ν) =
∫ ∞
−∞
· · ·∫ ∞
−∞
ejP
nk=1 νkαkfX(α1, . . . , αn)dα1 . . . dαn .
., '-+1X
2) . ('5.33 4 (/ 6. (;-;83 .-84 :(; 2) .-+1-1 5'3 3-' (; .'1.3 4 (-+5 ,-3φX(−ν) 72(2) . ('5.33 4 (/ ., *-'-+1
ν ∈ Rn 27 '(5
φX(ν)3-84:(;3 -7' 3/7 (3 ,22 (;33 ((75 6
φX(ν)., .. .-+11 5' 37 (;33 3-' (; .'1.3 -+11 +/3 3'415 (17 2(4 -) (. (,1 .,9 6X
-,'4,3 '(04 (3-9, *--(2. -.25X
2) *-5-7'3 27 *,) .(,'2 24 6(2-;3 .-84:(;φX(ν) =
n∏
i=1
φXi(νi)
6 32;717 * ()'2 .-: . -:--;,3 3-84:(;3 .,) * ()1
φX(ν) = E
n∏
i=1
ejXiνi
(8--2 .:.: . -:-; (,3 3-84 :(;3 *,) .(,'32 ');, ;32( 6. (2/(.3 .2;71 ,-3 32;713 .2/ (. .(2. -, ./. ((2-;3 .-84 :(;) 3,'. 37 (;33 3-' (; .'1.3) *()1 .,9 6* --(2. -.25 *:-3
X'(04 (3 -5-7' -9, 2 :7 32;717. (2/ (.3 .2;71 3(() 32;713 .2/ (.) 3+5(3 227 '+5) 52 *) 6. (-+11 +/3 . (-84 :(;3 2) .2;713 ,-3';1 '(5 .(2/ (.3 .2;712 3 (() 32;713 .2/(. '. (- 35'3 (. : ,7 , .-0 -00 .(2. -, .''( 3:-,6.(-2, -8::(;4, .(+/ (-1 . (-84 :(; ' (5 (:5 (
ν2) ' 272 +/, *--(0 -5 2) 2(+
'(5 * 3'( 3/ (: ..2 .-:) '5.1 6-:)3 ( (),'3 0:1 (13 ., .-',:-2 3-81'(; :'0 3:)1 +8-7 (:-,'6. -:--;,3 3-84:(;3) 52 *-)
φY (ν)2 '1 (2 27 (: 31 6
Y = AX.'7 (13 3-81'(; :'03 ., 85: 6
u ∈ Rn , φX(u)
(+- 32,)3:() +1-11 . (-32 * -2(7 -
X,Y
35().φY (ν) = E ejY T ν = E ej(A X)T ν = E ejXT A T ν = φX(A T ν)
X1, X2
X1, X2, X3 FX1,X2,X3(a1, a2, a3)
1.3
FX1,X2(a1, a2) = FX1,X2,X3(a1, a2,∞)
Y = AX
A =
1 0 0
0 1 0
; n = 3
φY (ν1, ν2) = φX
1 00 10 0
ν1
ν2
= φX(ν1, ν2, 0)
6νi ≡ 0
(5 * -5-81 (φX(ν)
., *-/4 (2 -9,Xi
(7 3:.)1 *-232 * -8(' *, -227 ;(,5 3:41
1.4
Y = aTX = a1X1 + · · · anXn = (a1, . . . , an)
X1Xn
ν Y
φY (ν) = φX
a1an
ν
= φX
a1νanν
a1 = a2 = · · · = an = 1 X
φY (ν) = φX1(ν)φX2 (ν) · · · φXn(ν)
6. (-:-; (,3 . (-84 :(;3 .2;71 ,-3 . 5 , 1 *(7 2) .-:-; (,3 3-84 :(;3 3:41'5(+1) ((-7 ,7 6* -:.)1 ';1 2) .-:--;, 3-84 :(; (:25-4 *+(43 3'415 ( 1.2)
1 3:() (9 3,8(. -7 52 * -) :6-'24 3:.)1 2) .-:--;, 3-84:(; (:254 -'24 3:.)15 72(*(75254:( '(2--0 '(02 * -0 ::(;4,3 ., /.;: 6, 1
Y,X(-3 - 3'3
E[
ejXν1+jY ν2
]
= E
∞∑
m=0
(jν1)m
m!Xm ·
∞∑
k=0
(jν2)k
k!Y k
6*-1--4 *3) -,:.5 EXmY k *-0 :1 (13 ., 2542 27 (:
ν1, ν25 . (4 9/ '(02
φX,Y (ν1, ν2)., /.;: *, 72(3-84:(;3 (9 35-1 6* -:.)13 -:) -;2
φX,Y (ν1, ν2)2) 3'-9 -+- 2 (2,7 * -0 :1 (1 2542 .-: -) 1 ;(,56* -;. ()1 *-3 (5 *-0 :1 (1 5 () -/2 / (: -277 .)1)1 .-:--;,3
1.5
X,N
Y = X +N
N X Y
X
Y N
deconvolution
φY (ν) = φX(ν) · φN (ν) ⇒ φX(ν) =φY (ν)
φN (ν).
Y µ
N X 1.6 λ
.(-5 - ( '(5 *-25(41 *3 *-- (, *-2+(1 .-),' 6. (5- ';11 21)/ .+:35 +/(-15 5()/ - (, 3 (2-;3(5 * -)1.)1 72( *-- (, *-(2-; * *-5 () -/ 852 .-/- 24 .-:) 63'453 ( .'()4.3 -1 (/.5 0';5 *-5'(2-;3 -7 (:2 3,'1 -97'13 2(53 0;)1 (52 6-:) ( (),' 0 :1 (15 4' * -:--:.1 (:, '),7 7 ( /(: 5 ('-476* -:04 *-:.)1 2) 2(+ (,2 5 (0 5 ('-4 3((31 - (, 3
-'243 3'413- .'+(1 .-+1-1 +/3 .- (, 3 . (;-;83 '(797 6- (, , 1
X,3-
fX(α) =1√
2πσ2e−
(α−m)2
2σ2 6
E[X ] = m, Var(X) = σ2
- 3:(. : - (, , 1 2) 3/7 (3 ,22 .-:-;(,3 3-84 :(;3 (φX(ν) = ejmν− 1
2 σ2ν2 6 Y = aX + b
'(5 6X ≡ m '1 (27
σ2 = 0((:13 3'412 (:5 31
φY (u) = E ejaXu+jbu = ejbuejamu− 12a2σ2u2
= eju(b+am)− 12a2σ2u2
(.2(σ2 = 0)
*-:((:13 , 13 27 .2273 - *- (, 3 *-,'4,3 *-:.)13 ./;)1 ., 5-/'32 * -7 : *, (.; (35 '(8 ,22 - (, , 1 +-1. ,-3 - (, , 1 2) .-',:-2 3-81'(; :'0) 3,8(.3 ., 254: -9, 3/;)13 ('85
1.3 -;2 - (, 3 , 13 ., '-+: 72( (9 32273 . () 2 -,+7 ( / (: )135 3,':) -;7 6.(+/ (-1 . () -'+6
1.4 -;2 . ('-) - (, ((:13 3'413
31 (+ '5+3 6. (;-;83 '+ '),1 3-55 '. (- '-) - 2(;-0 ');,1 .-:--;, 3-84 :(;2 '513 *-.-2) *-, (' (:,6. (. (, / (. -: -,) (:5 3 -' (; .'1.35 ) (1-)2-'(04 (3 3'413
-,'4,3 3:.)13 a-+1-1
n-,'4, ,2 '(04 ( 27 '(5 *, - (, ,'4: -+1-1
n X, ( -,'4, '(04 ( 3'+327 *, - (, ,(3 , () +-32 27 (: 9, 6'5+ 3:) - ,2 39 |a| = 1
) ('+2 * ');, 6- (, , 1 , (3aTX = (a,X)6- (, , 1 , (3 ((7 272 (2) 372)3
,-532 ');, ;33 * 31 6- (, , 1 ,(3 5-7' 27 72( 372)3 , (3 (2) 5-7' 27 -9, - (, , (3X
, ( *,(::-, -,'4,3 '(04 (3 *2(, *-- (, *3 *-5-7'3 27 .1-(1 . (0 :-+' (, (4 .7'15) -,'4, '(04 ( 2) 31 (+
'(04 ( , (3 '(04 (3) 5--/1 ,2 --+ 39 * -- (, (-5-7' 27) -,'4, '(04 ( (. : *, 72( - (, -,'4, '(04 (6- (, -,'4,Y
.= X ·S S = ±1
X
S X
1.7
EY = 0
(X,Y ) 1/2 0
X + Y
X,Y E[XY ] = 0
*-- (, *--,'4, *-'(04 ( 2) . (:(7.6- (, , (
Y = AX* - (, , (
X + b* -9, - (, , (
X*, 3:0 6, ./7(3
(a, Y ) = (a,AX) = aTAX = (A T a)TX
Y72( 5 (17 ((-7 (. (,5 /'735 ,2
X2) 372)3 '5+ 2) (;(5 ,-3 * -(1 ((-75
Y2) 372)3 27 72(6- (, , (
7n× n .-2-2) -, ( . -'01-
Λ38-'01 (
m-+1 -1
n'(04 ( *--4 *, 4'( *, -+1-1
n- (, , (
X3:0 65
- 3:(. :X
2) .-:-; (,3 3-84 :(;3)φX(ν) = ej(νT m)− 1
2 νT Λ ν
= ejP
νimi−12
P
i
P
k νiνkλik
*--4.1 9, (EX = m
Λ = E(X − EX)(X − EX)T = [λij ]i,j =[
Cov(Xi, Xj)]
i,j
5 :, -'((43 .8-'01 ., ( *-8(113 '(04 (( ., 1 : 6- (, '(04 (( , (3X
) /-:: 3/7 (3m
.= E[X ]
Λ.= E
[
(X −m) (X −m)T]
- (, '(04 (( .'+31 6Y = νTX
)+/ , 1 '-+:(ν'(04 (( '/5: 6
Λ.= λiji,j 5 38-'013 -'5, ., 1 :, (3 (2) 8(113 -7 5(: .2/ (.3 . (-',:-21 6- (, , 1
Y-7 5(:
mY = νTm
, (3Y
2) :, -' ((3) 254: * - :, -' ((4 2) *+(4 5 () -/1 (σ2
Y = νT Λ ν =
n∑
i,j=1
νiνjλij
,-313+(4 :5 (2) .-:--;,3 3-84 :(;3 - (, , 1
Y) ((7
φY (1) = E[
ejY]
= ejνT m− 12νT Λ ν
Y
.'+31 *2(,E[
ejY]
= E
[
ejP
ni=1 νiXi
]
= E
[
ejνT X]
.= φX(ν)38-'013(
m'(04 ((3 '),7 ) ('+7 . -:--;, 3-84:(; ) - - (, '(04 ((2 -7 (:/7 (3
ν272 (7 : 5 () -/3) ((-76 :, -' ((43 .8-'01( *-8(113 '(04 (( 4 (-+5 *3 .-:--;,3 3-84 :(;5 * - -; (13
Λ'/5: 6Λ
38-'013(m
'(04 ((3 * -:(. : 63:(. :3 3'(83 ., ) - .-:--;,3 3-84:(;2) /-:: -:)3 ((-73 ./7 (325() '-+:(a'(04 ((
Y = aTX.
.-:--;,3 3-84 :(;3 . (:(7.1φY (u) = φX(a · u)
= φX([a1 · u, a2 · u, . . . an · u]T )
= ej(aT m)u− 12 u2(aT Λ a)
aT Λa
.(:() (aTm
8(11 * - (, , 1 ,(3Y
-7 5 (: ,71 6X
2 3/:331 5(: ('/,3 (-(()3 '),7'(5 -7 *-+(- (:, (),'3 ((73 ./7 (31 6- (, -,'4, '(04 ( , (3X
3'+331 -'3 a272 (7 : '5+3) ((-7 (.8-'01 ( * -8(113 '(04 (( 4 (-+5 *3 .-:--;,3 3-84:(;5 * - -; (13
Λ38-'013(
m'(04 ((3 - (, '(04 ((6 :, -' ((43
1)1 -9, 6i27 '(5
δii = 1(i 6= k
'(5δik = 0
'),7 λik = σ2
i δik.-:(72, 38-'01
Λ) /-:: 6
X (:5 *, 6
Cov(Xi, Xk) = 0, i 6= k'1 (27 .((95 .-',:-2 *--(2. -.25
X1, X2, · · · *--,'4,3 *-:.)13)-9, - (, , (φX(ν) = ej
P
νkmk−12
P
λkkν2k =
n∏
k=1
ejνkmk−12λkkν2
k
* .1--4.1 - (, 3 3'415 *2(, . ((95 .-',:-2 . (2. -, .1--4 *, 4'( *, .-:(72,Λ
) (7 : +-1. 72(6. -0 -00 .(2. -, .''( .-',-:-2 . (2. -, - (, -,'4, '(04 ( ' (5 72( .-0 -00 .(2. -,
.,9 .1(2 6- (, , ( , (3X
-9, - (, , 1 , (3 *31 +/, 27 ( .5 *3 (-5-7') , ( , (3X
*, 3'3'(04 ( '1 (27 - (, (:-, '(04 (3 , - (, , (3 *31 +/, 27 3-82'(4 -'/ *31.7
31 (+5)X,Y
*-:.)13'(04 (( /'735 (:-, * -- (, (-5-7') '(04 ( *2(, .5 *-5-7'3 /'735 3-82'(4 -'/ (-5-7') - (, 63 -82' (4 -'/ (-5 -7' *, (2-;, - (,
*-'. (:3 * -5-7'3(n−m)
5 * --(2. -.25 -+1-1 n- (, -,'4, '(04 (( 2) *-:(),'3 *-5-7'3
m-.1 32,) 6+
3'(83 :,-' ((43 .8-'012 '),7 3/7 (3 ,22 35().Λ =
(
m×m0
0
(n−m)× (n−m)
)
(;33 ((75 31 6* --(2. *3Y
2) *-5-7'3 227 '+5 -9,Y = AX
. 5 * -5-7' - (, , (X
*, 32,) 63* -5-7'3 *, -7 . (5-)/ )- 72 .-',:-2 3-81'(; :'0 -+- 2 .5 (-5-7' '), '(04 ((2 -32 .-: *,36* -5 () -/3 . (-7 (5- ., 3( 3-53 ., 3 *-0);1 75 ( +';:5 5-7' 275 2;02 .-: . (5' . (-55 -9, .5- 3:(. : 35 ().3-9, 5()/ ,2 227 39 25, ;, .2/ (. 2) 3'415 ,7 (+: . (0); *)2 -+1-1
n- (, , (
X*, 3:0
.-0 -00 *--(2. -.25Y
2) *-5-7'3 (Y = DX
) 7D
32 ,'4: n× n .-'2(:- ,2 38-'01 .1--425 3-3-
Y2) -,'4,3 42/3 -) 1 ;(,5) 7 , 1 ,2( *- (54 (-3-
Y2) *-5-7'31 42/) 7.- 3'3
- '+ (1Y
( - (, , 1X1
'),7X =
(
X1
X1
) 2)12 6X
2) +1-131 )11 04 +1-1Y =
1√2
1 1
−1 1
X1
X1
=1
2
√2√
2
−√
2√
2
X1
X1
=
√2X1
0
1 : 6* --(2. -.25Y
2) *-5-7'3 -9,D =
1√2
1 1
−1 1
; detD = 1
.1--41D
) 52 * -)DD T =
1
2
1 1
−1 1
1 −1
1 1
=
1 0
0 1
= D TD
6D T = D −1 72(.1--4 -9, .-2-2) ,2 .-'01- 38-'01
Λ*, ,53 0;)13 ., 3/7 (3 ,22 254: 3:03 ./7(3 '(82'1 (27 .-:(72, ,-3
DΛD T 38-'013) 7 .-'2(:- ,2 5(17 72(D T = D −1 .-'0 -:(, .8-'01
DΛD T =
c1 . . . 0666c2
6660 . . . cn
6
.-',:-2 . 5 *-5-7'Y = DX
(.1 254.13Y-,'4,3 '(04 (2) /-7 (: 39 -:;2 *2(,
1.5 5 -- :( '(9/: +-1
EX = 0
'(79 E[Y Y T ] = E[DXXT D T ] = DΛD T =
c1 . . . 0666c2
6660 . . . cn
-.25Y
2) *-5-7'3 72( .-0 -00 .(2. -, .''( .-',:-2 . (2. -, 72( . (- (, 3 ./:3 ., - (: .7'. (- -227 ;(,5 6* --(2.
E ejνT Y = E ejνT D X = ejνT D m− 12νT D Λ D T ν = ejνT D m− 1
2
P
iν2
i ci
6;,1 3:()X
2) .2/ (.3 *, * *--(2. -.25Y
2) *-5-7'3 72(di
'),7D T = (d1, d2, . . . , dn)
1 : 3,53 3'35 4;. : D
38-'013 ., *-,8(1 -, 1.5
2 '(9/:* ()'2 27 (: -9, 6-+1-1
n'(04 (
ΛD T = (Λ d1,Λ d2, . . . ,Λ dn)
D T
c1 . . . 0666c2
6660 . . . cn
= (c1d1, c2d2, . . . , cndn)
.-'0 -:(,D
( . (-3 1.5
(.1 5 (: 72ΛD T = D T
c1 . . . 0666c2
6660 . . . cn
*3ci
.('/, *-2-15 Λ d = cd
3, ()13 ., *-'. (;di
*-'(04 (3 72 6Λ di = cidi
254: . (+(1 .,(()3 - (6D T = D −1 *--4.1 * 72(
Λ2) *--183 *-'(04 (3 *3
di(
Λ2) *--183 *-7'3
- (. : - (, 3 -+1-1 n3 -2(3 (2-;3 -9, .-'2 (:- ,2
Λ*, 3/7 (3 ,22 6(
fX(a) =1
(2π)n/2(det Λ )1/2e−
12 (a−m)T Λ −1(a−m) =
1
(2π)n/2(detΛ )1/2e−
12
P P
(ai−mi)(aj−mj)θij
6Λ −1 = [θij ]
'),7)'(;1 5 () -/ '(82 6Λ38-'013 2) -7;(33 ., 5)/2 ) - .)'(;1 3'(85 . (;-;83 ., * ()'2 -+7 -7 52 * -)6) ('+ (:-, 397 5 () -/ .-:--;,3 3-84 :(;3 2)3-81'(; :0 (8-5 ( .5 *-5-7' * , +1-1 (. (,5 - (, , ( 2) . (;-;83 5() -/ -+- 2 -32 .-: (9 3/ (:26
Λ = AAT .1--413A
38-'01 -+- 2 .-',:-2./ (:5 9, ( 6- (, 7 * , (3
X2(. : '),7
X12) . ('5.33 4 (/ -9, - (, , (
Y =(
X1
X2
) *, 3/7 (3 ,22 692) .-:. (13 .2/ (.3 -;(. .-:. (13 .-:-; (,3 3-84 :(;3 ./ (:5 (,X2
(. :X1
2) 3:. (13 -2(3 (2-;363-:.35 .)'(;1 3'(85 -(2. (::-, 3:. (13 '(9-;3 39 3'415) ''5.1 63:. (13 '(9-;3 (X2
(. :X1
* (7-6. (-',:-2 . (-81'(; :'02 . -0 :-' (:-, . (- (, 3 .:(7. 6,
6*-:(),'3 *-0 :1 (13 -:) Λ ,m
- .-1)1 +/ 54:X
- (, , ( 2) (2-;3 656. -0 -00 .(2. -, .''( .-',:-2 . (2. -, 6
6* -'-83 .7'1 5(5 -+- 2 . 5 , 12 '(52 .-: 6+
,-3X
(. : '),7Y
2) 3:. (13 (2-;3 -9, +-+5 , 1X
/-:: , 1Y,X
FY |X(α|β) = PY ≤ α|X = β =PY ≤ α,X = β
PX = β
-8' , 1X
*,(FY |X(α|β) = lim
ε→0
PY ≤ α ; β < X ≤ β + εPβ < X ≤ β + ε 3:. (1 -2( (2-;
fY |X(α|β) =∂FY |X(α|β)
∂α=fY,X(α, β)
fX(β)-9, .(1 --4 . (-(;-;83 ( *--(2. -.25Y,X
'),7 +/ (-15(fY |X(α|β) = fY (α) fX|Y (β|α) = fX(β). .-:. (1 .2/ (.
E[Y |X = β] =
∫ ∞
−∞
αfY |X(α|β) · dα
'. (- -227 ;(,5(E[Y |X = β] =
∫ ∞
−∞
αFY |X(dα|β)
')43 *--4 39 3'415 * ) .(,'32 ');, 6β2) 3-84:(; , -3
X = β5 3:. (1
Y2) .2/ (.3) 52 5-)
E[g(Y )|X = β] =
∫ ∞
−∞
g(α)fY |X(α|β)dα
'. (- -227 ;(,5(E[g(Y )|X = β] =
∫ ∞
−∞
g(α)FY |X(dα|β)
-9, X = β
.:35Y
2) *-:. (13 .(;-;83 (, (2-;3 ., *-+(- (:, *, '1 (27 63-:.33 2) 3-84:(; (9 5 () (2) *-:. (13 .(;-;83 (, (2-;3 ., 5)/2 '(8 -,X = β
.:35Z = g(Y )
2) .-:. (13 .2/ (.3 5() -/ '(826X = β
.:35Z
) .(,'32 3)4 ,2 -9,Y = aZ1 + bZ2
*, 6. -',:-2 ,-3 .-:. (13 .2/ (.3 * 32-' .2/(. (17E[Y |X = β] = aE[Z1|X = β] + bE[Z2|X = β]6
E[Y |X = β] = E[Y ]*--4.1 *--(2. -.25
Y(X
'),7 7 (17 6β2) ' 272
-9, . 5N
(X
'),7Y = X +N
*,) ,71E[Y |X = β] = E[X |X = β] + E[N |X = β] = β + E[N ]3:.)13 ., .,9 3-84 :(;5 5-8: .7 *, 6
Ψ(β)5 .,9 3-84 :(; 1 :
β2) 3-84:(; , -3
E[Y |X = β]'(1,7
(:--3+E[Y |X ]
5 39 -,'4, 3:.)1 12 (3 : 6) +/ -,'4, 3:.)1Ψ(X)
3,8(.3 3-3. (18X
-,'4,36Ψ(β) = E[Y |X = β]
'),7E[Y |X ] = Ψ(X)6
E[Y |X ] = E[Y ]*--(2. -.25
Y(X
'),7 52 * -).-:. (1 .2/ (. 2) . (; (: . (:(7.
E[X |X = β] = β,
E[h(Y )g(X)|X = β] = g(β)E[h(Y )|X = β]5
E[h(X,Y )g(X)|X = β] = g(β)E[h(β, Y )|X = β]'. (- -227 ;(,5 (
E[Y ] = E[E[Y |X ]]
342/33 .:(7. 2) ,53 /( :3 5(: * 31 6342/33 . (:(7. . (,'4: ( 5 .(:(7. 3'3E[
h(X,Y )g(X)]
= E[
g(X)E[h(X,Y )|X ]]
2) 3/7 (3 -1
E[
g(X)E[h(X,Y )|X ]]
=
∫ ∞
−∞
g(β)[
∫ ∞
−∞
h(β, α)fY |X(α|β)dα]
fX(β)dβ
=
∫ ∞
−∞
∫ ∞
−∞
g(β)h(β, α)fY |X(α|β)fX(β)dαdβ
=
∫ ∞
−∞
∫ ∞
−∞
g(β)h(β, α)fY,X(α, β)dαdβ = E[
g(X)h(X,Y )]
-+1-1n'(04 ( *--4 -9,
EX = 0, EY = 0-+1-1
n+1 - (, -,'4, , (
(X,Y )*, 3/7 (3 ,22 35()/ 3,8(.
) 7a
E[X |Y ] = aTY
;, .2/ (. 2) 3)-'+3 ,22 - (, -,'4, '(04 ( , (3(X,Y )
*,) 5(: .,9 3,8(.1) 18 ., :7) 3'3) 7
a*--4 -9,
E[X |Y ] = E[X ] + aT (Y − E[Y ])6-+1 -1 +/ '(04 (Y
(5 3'413 2 5 ()/2 32-/. 3 : :7.)32 2 3)4 ( 3+-15
1.8
fX1,X2(x1, x2) =1
2πσ2√
1− ρ2e− 1
2σ2(1−ρ2)(x2
1−2ρx1x2+x22) .
m = 0, Λ = σ2
(
1 ρρ 1
)
.
x1
fX1(x1) =1
σ√
2πe−
x21
2σ2
fX2|X1(x2|x1) =
1
σ√
2π(1− ρ2)e− 1
2σ2(1−ρ2)(x2−ρx1)
2
σ2(1−ρ2)
ρx1
x2 x1
x1
19 ., (()3 -+- 2 ( *-:-((21 * -' (+-) 02(4 '-)713 (Global Positioning System) GPS
'-)71 2 (; +8-7'(+7 -:; 2 *-,81: (:, -7 (:2 '1 (, 4/'1 2) 39 (. : 6-((2 271 4/'13 ., /:;1 '(+-)3 192 /-5 30-243 (, *-'(+7 -:) 2) (. -/ ,-3) 3+(4:5 * -(,81: (:, .(+-+1 -.) (:2 ) - *, 6++1:3 4/'13 , (3 ( (-+'). (+-+1 -.) -7 . (,'2 24 -9, *-3 -:; 21 2)12 (:2) 35 (3 (+- *, 621 227 '+5 ',.1 . (+(4: 2) 39'(+7 , (3) (: ' (+7 * (. -/ .(: 35(3 * (4-13 ., /:;2 -+7 . (4 -;1 *-:-((2 -:)1 4/'1 2) . (4 --(+16. (+-+1 ) (2) . (4 -;1 35( 2 +-1 ,22 6&',3*-:-((21 4/'1 .(+-+1 .; (3 -+- 2 4 (-+3 ., *-';)1 +8-7 6. (12) (1 :-, . (+-+13 -)1 ;(,5 *2(,6 (')3 .--5 4 (-+5 -3 (9 * (413 2) 37'3 2542( *-:(. :3 27 ., 224)2 +8-7 6* -; (: (+- (:-, (1 (4 -1) '(42 ((7 54:
c(
ba.(+(4:3 ) (2)1 '(4 '(., 2) ,-3 '. (- 30 (); , 31(+ 31 (+6(. (, (542 * -8(' (
6 '(-,2) 3'415 (. -/3 .+(4 :7 * (4-13 ., *-5(4 (:--3
c1 ./, (
a1 ./, 2)12 . (+-+1 -.) 4' (:2 (-3 *,. ('8(- . (+-+13 .) (2) '),7 *2(, 391 *2.: '7 25, +-1. '. (-5 35(03 3 -543 3:-, (9 )' * . (+-+1'+ 3-3- 31 (:5 ( (')3 '(43 2) ' ()13 * (4 -13 ., (542 -, 32,)3 .2,) : (. -/ . (+(4 : ) (2)6 (')3 .,-) 2) 2+(3
.(4:2 (:-8' 35 . (,1 (+ (:-,'1.1
-5 6-(8' 2+( -5 2 3')3 (, 37'35 4 ()+/1 37'3 (') 63 9 ,) (:2 . (+( -3 ., /-:: 39 4';5 6 +7 ( . () (' . (+-+1 -; 2 3'012 4/'1 -'32 :2 *-)'1 . (, (')3 .-5 2) -227 / ( -:
+(+12 * -2(7 - (::-, (. (, X
-,'4,3 '(04 (3 ., ')2 *-8(' .(+-+13 '(04 ( Y
-,'4,3 '(04 (3 -(81 6,
5(0 , (3) 27 ')1 6(Y , X)
2) *-;. ()13 *- (2-;3 ,(3 (:-+-5) +-13 6, ( , (3(Y , X)
( 3'-) - 3'(856X = φ(Y )
5 1 : '), 3+-+13 2) 3-84 :(; , (3 ' (,Y
(.1X
., -'32 30 -) , (812 * -8(' (:--3 6, ( ,2( , 1 , (3X
) 3'415 32-/. 4 ( : . (0); *)2.-21'(; 3'(85 6(3)27 5(15
X2 5 ('4 3-3-
φ(Y )) 7
Y2)
φ3-84:(; , (812 '1 (27
3,-)3, E[g(X,φ(Y ))]
'-+: φ(Y )
')1 27 '(5 g(·, ·) 3,-)2 ++1 .:35 5-0 (-'0 -'4 .'+3 6531 (+ 6
E[(X − φ(Y ))2],-3 .8(113 3,-)3
g(a, b) = (a − b)2 '(5 31 (+2 6g
(-'0 -'43 -;2 .8(113.'()4.5 6E[|X − φ(Y )|] ,-3 .8(113 3,-)3
g(a, b) = |a− b| 3,-)3 2) 02/ (13 '3 .2/ (. .'/,g(x, y) = 0
*-'/(5 (7 : (')2 -5'1 -(7 - 2542 -+7 (0, 1
*-7'3 ., 2541X
-(8'3 . (,3 2)12 .-.';6x 6= y
*,g(x, y) = 1
(x = y
*,3-3. .8(113 3,-)3) 7 *-:.)1
n2) 3-84 :(;
φ(·) ,81g5-0 (-'0 -'4 2 .02/3) '/,2 3-53 66. -21-:-1
(, PX,Y
. ()13 (2-;3 (+- .(,53 . (+(4 :3 -.) 2 +-;432 ) - (93 3) -3 ., 2-;32 3-3- .-:) -+73,-) (-'0 -'4 *-5(4 PY |X
3:. (13 (2-;3 7 (PX (a-priori)
32-/.721 . ('5.3 (2-; * - (+- -;(2-/2')13 2) .8(113 3,-)3 ., *-5)/1 71 '/,2 63 9 (-'0 -'4 -; 2 -21-0;(, ')1 , (812 *-;, () 9, (-;2 .8(11 ,-3 ,2, *-+(- (::-, (. (, -(8'3 2+(3 2) -. -1, '2 /-5 3:-, 3,-)3) 52 * -) : 6,81:)6* --');,3 *-7'3 (2-;(2-;3 -;2 '(43 -1 (4 -1 27 ; .8(11 ,2, '(43 * (4-15 3-(2. 3:-, .8(113 3,-)3 '(43 2) 31 (+5632-/.721.-:('4 3+(4 : .1--4 -5 2 6* -:() *--21 -0;(, *-7')12 (2-5 (- . (:() (-'0 -'4 . ('-/5 -7 52 * -) : -5 2.72 57 (7 32.3 2)12 63'41 275 7 '5+3 -, *2(, 32-/.721 (2-;
X2 /-2 27 (:) -:(-3 *-'41 35'35
+-,1 39 57 (72 (:.,1 4/'13 2) 32-/.721 (2-;2 /-2 .-: 5 (1 39-, 6(-2, 4/'13 ., +(+12 *-8('( ) +/. ('5.3 * (-42 5(1 ) -) *-/-:1 (:, '),7 6-' (-';,3 (2-;2 5(1 ) - .'()4.5 . (-5 (7 .('/, . (-556Bayes
4 (/ ) .-:, - --5 3 -5 .,'4: 3-53 32-/.721
.('5.3 (2-; * --4 -(81Y
-(8'X
, ( ,2( , 1 , (3Y
(5 3'415 97'.: , (515 3 8(3) (')3 .-52 '(9/:* -'-+1 3.'95(g(·, ·) 5-0 .';- *-5(4 6
X = φ(Y )5 1(1
Y++1: '),7
X2 ')13 6
X(Y
2 . ()12+(3g(a, b) = (a − b)2
.8(113 .- (5 -'3 3,-)3 (-'0 -'4 '(5 6E[g(X,φ(Y ))]
.8(113 3,-)3 .,.- (5 -'3 3,-)3 , (3E[ε2]
( . - (5-'3 3,-)3 , (3ε2
(')3 .,-) ,'4:ε , X − φ(Y )
7 '+(13ε'/5: 3,23 ( 3.1 6.8(113 3,-)2 3-89-1 -:-1 3) ()
φ0(·) ., *-);/1 3'41 275 6')13 2) .8(1133,-)3 (-'0 -'4 39 3-3- )'(;15 (-'0 -'42 /-.: ,2) * (41 275 ( .8(113 .- (5-'3 3,-)3 (-'0 -'456.8(113 .- (5 -'3
φ(·)
E[(X − φ0(Y ))2] φ0(·) 2.1
Xopt = φ0(Y ) = E[X |Y ].
6Y
3+-+13 ' ., 3254.3) 3-84:(;5 *-5-81 (E[X |Y = β]
., *-5)/1) 5(17 ,-3 . (1)136)'(;15 -21-0;(,3 ')13 5) (/1 35) .(,1 (+ ) (2) . 83 '/,2 +-1 ) -: (9 3:0 ./7(32
.-:70 ,1 (+ *. , ,1 (+fX,Y (α, β) = α+ β ; 0 ≤ α, β ≤ 1
fY (β) =
∫ 1
0
(α+ β)dα =1
2+ β
E(X |Y = β) =
∫ 1
0
αα + β12 + β
dα =13 + β
212 + β
6E[X |Y ]
., 5)/: 652()15 *- (, *--,'4, *-:.)1X,Y
5 ,1 (+1 :
Xc = X − E[X ] ; Yc = Y − E[Y ]
142+7 -,'4, 3:.)1 3:5:(Z.= Xc −
Cov(Y,X)
Var(Y )Yc
. ()15 *-- (, Yc
(Z
) .(,'2 24 .()15 . (- (, .'+31 6E[ZYc] = 0
(E[Z] = 0
) '('5Z.'+3 (.1
72( E[Z|Y ] = E[Z] = 0
( * --(2. -.25Y, Z
* ) ,71 6* --(2. -.25 *3) 5(: 3-82'(43 ' (/1 (0 = E
[
Xc −Cov(Y,X)
Var(Y )Yc
∣
∣
∣
∣
Y
]
6 = E[X |Y ]− E[X ]− Cov(Y,X)
Var(Y )(Y − E[Y ])
6 ,71
E[X |Y ] = E[X ] +Cov(Y,X)
Var(Y )(Y − E[Y ])
-+- 2 (. :Y
(.1X
2) -21-0;(,3 ')13 72(φ0(β) = E[X ] +
Cov(Y,X)
Var(Y )(β − E[Y ])
6 Xopt = E[X ] +
Cov(Y,X)
Var(Y )(Y − E[Y ]).
6
.(:('/,3 . ('()3 * 3(()3 6. (+-+13 2) .-',:-2 3-84:(; , (3 39 3'415 -21-0;(,3 ')13) 72 52 * -)X,N
'),7Y = X + N
(5 5 ()/3 3'415 (:5 6E[X ]
8(11 * - (, , 1 , (3Xopt
72 61.6
- 2)6X
2) :,-' ((5 4' -(2.Xopt
2) :,-' ((3 -7 . (,'2 24 3-82'(4 -'/
.-'(04 ( 3+-+1 2) 3'412 322736E(X |Y )
., 5)/: 6. -'2(:- ,2 38-'01 ,-3Λ
Y= CovY -7 /-::( - (, , (
(Y ,X)Y = (Y1, · · · , Yn)T'-+:
Y c = Y − E Y
Xc = X − EX
Z.= Xc − Y T
c Λ−1Y E [Xc Y c] .6. 5( .()15 *-- (,
Z, Y c72 6
EY cZ = 07 (
E Z = 0-9,6
E(Z|Y ) = EZ = 0,71
'1 (27E(Xc|Y ) = E
[
Y Tc Λ−1
Y E(XcY c)|Y]
= Y Tc Λ−1
Y EXc Y c . '. (- .)'(;1 3'(85E[X |Y ] = EX + (Y − EY )T Λ−1
Y
Cov(Y1, X)666Cov(Yn, X)
6
(X,Y ) 2.2
E [X | Y ]
E [X | Y ] = E[X] +[
(VarY )−1 Cov(Y ,X)]T
[Y − EY ]6
5 . (:1 (13 . (,8(.3 22(7 3. (, '(792 5 ()/( 35 ()/ *2(, 30 (); ,1 (+ ,1 (+Y = X +N
6 E[X ] = E[N ] = 0
6 E[X2] = 1, E[N2] = σ2
n6 ., ')2 *-8(' ( ) '5 2(50 2:-3 ., '1(27
Y., *-++(1 ) '3
N2:-3 , (3
X . 5 * -- (,
X,N6X
-4 :3 2:-3. (,8(.5 )1.) : ,2 .,9 31 (+ '(82 6;, .2/ (. .()15 *-- (, *3 72( . 5 * -- (,
X,N) (:/:3
*()'2 27 (: 72 61.6
- (5 3/7 (3 ,22 -; (3) 0;)15 ,2, 5 ,1 (+E[X |Y ] = c0Y
.8(113 .- (5'3 3,-)3 2 3-89-1 -:-1 85: c0
., , (812 .:1 2 6c0
(+- ,2 (54 '(5
E[
(X − cY )2]
= E[
(X − cX − cN)2]
= (1− c)2 + c2σ2n
3'-/5 -+- 2 X
-(:-) -, -5 c0
.1,.3 -+- 2 3'); .);/1 3-89-1 -:-13) -'3X = c0(X + N)
) ((-72 3, (()3 ( 3'-9 - .:. (: 3-89-1-:-1 6+-,1
c0 = 03'-/5 -+- 2 ) '3 4 (2- -52 +/1
c0 = 1
c0 =1
1 + σ2n
254:σ2
x,(3
X2) :, -'((3 (5 '. (- -2273 3'415
σ2x
σ2x + σ2
n
6 5(03 (')3) 3,':
( 2.7)2 (),' 0515 6) '3 ( . (,3 2) . (-/-3 .(18(5 -(2.3 2 -5 ';1 , (3)
04σn
37 (1: ) '3 .18( '),7 '. (- . -:(-3 -:) 0515( .'/, 3,8(. . (: * 25,
Y, (3
X2) '. (-5. (1)1 .'/ 3+-+13 2(+
σn 33(5 )'3 .18( '),7 , 3+-+13 .1,5 , (3
X2 '. (-5 5 (03 ')136;, -/7 (:3 3'415 (2) 8(113 , (3
X2 '. (-5 5 (03 ')13 72( 3) 12
,-3 X
2) -227 :,-' (( 2) 3'415 .'. (:3 3,-)3E[
(X − c0Y )2]
= E[
(X − c0[X +N ])2]6
=
(
1− σ2x
σ2x + σ2
n
)2
σ2x +
(
σ2x
σ2x + σ2
n
)2
σ2n
6 =
(
σ2xσ
2n
σ2x + σ2
n
)
62.1
3:0 ./7(32 '(9/:, (3) 27 ')1 '(5 6
E(φ(Y ) −X)2 ≥ E(φ0(Y )−X)2*--4.1
φ, (3) 27 ')12) . (,'32 )'+: 3/7 (3
*--4.1 Y
(. : '),7X
2) .-:. (13 .2/(.3 ,-3φ0(Y )
'),7 φ(·)
E[
(X − φ(Y ))2]
= E[
(X − φ0(Y ) + φ0(Y )− φ(Y ))2]
= E[
(X − φ0(Y ))2]
+ E[
(φ0(Y )− φ(Y ))2]
+ 2E[
(X − φ0(Y ))(φ0(Y )− φ(Y ))]
2 .2/(. 7 /, ( Y
5 3:. (1 .-:. (1 .2/ (. *+(4 85: E[(X − φ0(Y ))(φ0(Y ) − φ(Y ))]
('/,3 '5-,5 --:254 : ( 5 -;2 6Y
E[
(X − φ0(Y ))(φ0(Y )− φ(Y ))]
= E[
(φo(Y )− φ(Y ))E[X − φ0(Y )|Y ]]
*--4.1 (E[
X − φ0(Y )|Y]
= E[X |Y ]− φ0(Y ) = φ0(Y )− φ0(Y ) = 0
72(E[(φ0(Y )− φ(Y ))2] ≥ 0
(:5 6;,.1 -) -2)3 '5-,3 '1(27E[
(X − φ(Y ))2]
≥ E[
(X − φ0(Y ))2]
(3 (17 5 (0 ) - -2(, .8(113 .- (5 -'3 3,-)3 (-'0 -'4 -; 2 '. (-5 5(03 ')13 , (3φ0(Y )
) '('5 ,71(6(:11 5(0 -, *2(,6Y
, ( 3 -3-Y
, 1 * (415 *, * -(:-) ,22 .',) : 3,8(.3) 52 *-) 63:03 ./7(3 ., (:12)3 75-9, -3)27 3-84 :(;
g-3. 3'3
E[
(X − φ0(Y ))g(Y )]
= 0 .6 .(:(7.1 2-2 3/7 (32 31(+ 3/7 (33 6. (+-+13 2) 3-84:(; 27 * 3-82'(4 .'/ (')3 .,-) ,-3 . (1)13254: ( 5
E[
(X − φ0(Y ))g(Y )]
= E[
(E[X − φ0(Y )|Y ])g(Y )]
= 0
6φ0(Y )
.'+31 , (3 ('/,3 (-(()3 '),76+-/- , (3 -21 -0;(,3 ')13 -7 /-7 (32 27 (: 3'33 .'95
φ1
φ0 2.3
E[
(φ0(Y )− φ1(Y ))2]
= 0
φ0
3/7 (3E[
(X − φ1(Y ))2]
= E[
(X − φ0(Y ) + [φ0(Y )− φ1(Y )])2]
6 = E
[
(X − φ0(Y ))2]
+ E[
(φ0(Y )− φ1(Y ))2]
+ 2E[
(X − φ0(Y ))(φ0(Y )− φ1(Y ))]
.6
-7 (:25-4 3,-) 3.(, * -7')13 -:)2 3/:33 -;2) ((-7 6;,2 3 (() ('/,3 '5-,3 2-2 3'33 225E[
(φ0(Y )− φ1(Y ))2]
= 0
63 +-+1 27 '(5 3,8(.3 3. (, ., *-:. (: *3) 5(15 *-39 *3 *-7')13 72(6-0 (2/2 339 .-' (04 (( 3+-+1 '(5 3/7 (33
:(5.: (9 3+(4 : -532 -+7 6')13 2) 3:512 '),5 ,2( (')3 2) 3,8(.2 '),5 ,-3 . (+-/-3 -7 52 * -) :φ0(Y ) = Y1, φ1(Y ) =
*-7')13 -:)) 5 (17 6X = Y1 = Y2
. (+-+1 -.) 1 2 (') 2) '. (-5 0 (); 3'41563:() ')13 '(5 3/ (:3) . ('12X
,-3 (')3 .,8(. *-'413 -:)5 *--21-0;(, *3Y2.4-'9 .,8(. 2) (')5 .4 ( (9 ,1 (+ 6-21-0;(,3 ')13 2) )'(;1 5 () -/2 .; (: ,1 (+2 .7 '(5:6. -21 -:-13 . - (5'3 3,-)3 (-'0 -'4 -;2
i = 1, · · · , 6 pi = 1/6
.:(3 3-5 (4.'. (:3 3,-)3(
E[X ] = 3.5'1 (27 8(113 , (3 -21 -0;(,3 ')13 . (+-+1 ,22 6,
E[
(X − EX)2]
= E[X2]−(
E[X ])2
=1
6(1 + 4 + 9 + 16 + 25 + 36)−
(
31
2
)2
=91
6−(
31
2
)2
= 2.72
3)2)31 +/,X
) (, 3)2)31 +/, ,-3X
3,8(.3 *, (:2 ';1 (3) -1 3-81'(;:-, * -2541 (:, 65142+7
Y, 1 '-+: 6
Y = 0 if X = 4 or 5 or 6
Y = 1 if X = 1 or 2 or 3
*--4.1PX = 1|Y = 0 = PX = 2|Y = 0 = PX = 3|Y = 0 = 0
PX = 4|Y = 0 = PX = 5|Y = 0 = PX = 6|Y = 0 = 1/3
7 (17 6E[X |Y = 0] = (4 + 5 + 6)1/3 = 5
72PX = 1|Y = 1 = PX = 2|Y = 1 = PX = 3|Y = 1 = 1/3
PX = 4|Y = 1 = PX = 5|Y = 1 = PX = 6|Y = 1 = 0
3-3. .'. (:3 3,-)3 6Xopt = 5
-9,Y = 0
*,(Xopt = 2
-9,Y = 1
*, * (7-2 6E[X |Y = 1] = 2
72E[ε2] = E
[
(X − E[X |Y ])2]
= E[X2]− E[
(E[X |Y ])2]
=91
6− 1
2(52 + 22) =
2
36+-1 ,22 3-3 (')3 '),7 (:25-4 '), 3,-)31 .-. (1)1 3:04 3,-)E[X |Y = 0] = 16/3
.- 31(+ 5 () -/ -9,0, 2, 3, 4, 5, 7
(-3 - * -7'3 2)12 3-5(43 2) *-7'3 ., 3:) : *,72 35-3 6
P(Xopt = X) = 00';5 63-5 (43 2) *-7' *:-,) *-7' (:25-4 '1(27
E[X |Y = 1] = 5/37 (6(-((-) . )32 *,. (1 (:-, , 3:04 .8(11 .- (5-' 3,-)2 , (+ .- (5 -'3 3,-)3 (-'0 -'4) ,-3
.()13 . ('5.33 4 (/ 2) +- 39 5 () -/ )'(+ 3'41 275 6'. (-5 3)4 .-:. (13 .2/ (.3 5() -/ *-5' * -'415')13 ., , (812 .-: ,2 -');, (:-, . -:. (13 .2/ (.3 5 () -/ '),7 (, (+- (:-, 39 4 (/ '),7 6
X(Y
2)
97'.: 232 65 () -/2 .-: , .(/; 5(0 ')15 227 '+5 * -4;.1 32,7 * -'415 6Y
.:-35X
2) -21 -0;(,3,( - (, '(04 (( , (3
Y ,X) *-/-:1 (::-, 39 -5 6. (+-+13 2) .-',:-2 3-84 :(; *3) *-7')1 ./;)15*-;. ()13 *-0 :1 (13 ( (),' '+1 *-0 :1 (13 . -+-5 4;. : (:, 6*32) .()13 (2-;3 (+-) /-:32 '08: ,26-:) '+1
-'243 3'413')12 39 3'415 6./, 3+-+1 1 2 85.1 (')3 '1(27 , ( ,2 ( , 1 , (3
Y(5) 0 ();3 3'413 1 2-/.:
3,53 3'(83 ., -227 ;(,5 ) -Y
2) .-',:-2 3-84 :(; , (3)X l = aY + b 6*3)27 * - (54 *3
b(a'),7
142+7 '-+: -21-0;(,3 -',:-23 ')13 .,-81 .-5 .,.('/, *-2-15 6. -21-:-1 3-3. .8(113 . - (5-'3 (')3 .,-)) 7
X l = aY +b3'(831 ')1 ,81 3-5-(0 -53) 7
b(a*- (54 ,81
E[ε2] = E[
(X − X l)2]
= E[
(X − aY − b)2]
*- (54 -:) , (812 ) - ,7 -7 -21 -0;(,3 ')13 .,-811 30 (); '. (- 35'3 3-5 (9) 52 *-) 6-21 -:-1 3-3 -6X = φ(Y )
312) 3-84 :(; ,2( +525.8(113 . - (5-'3 (')3 .,-)2 -(0 -53 ., .; (: *; * ()': ('.;
E[ε2] = E[
(X − aY − b)2]
= E[X2]− 2aE[XY ]− 2bE[X ] + a2E[Y 2] + 2abE[Y ] + b2
*--42 -'8 b∗
-21-0;(,3b) 254: 2 .'9:3 ., 3(() :(
b-;2 2-2 -(0 -53 ., '(9: *,
b∗ = E[X ]− aE[Y ]
254: .8(113 . - (5-'3 (')3 .,-)2 -(0 -53 (.2b∗
.5835E[ε2] = E
[
(Xc − aYc)2]
= E[X2c ]− 2aE[XcYc] + a2E[Y 2
c ]
-21 -0;(,3a) 254: 2 3, (()3 (
a-;2 2-2 -(0 -53 .'-9 - 5() 6
Xc = X − E[X ](Yc = Y − E[Y ]
'),7*--42 -'8
a∗
a∗ =E[XcYc]
E[Y 2c ]
=Cov(Y,X)
Var(Y )
, (3 -21 -0;(,3 -',:-23 ')13 72(X l
opt = E[X ] +Cov(Y,X)
Var(Y )
[
Y − E[Y ]]
.254.13 .8(113 .- (5 -'3 3,-)3 6)135 1 3'3 3,' 6 -5 5 31 (+2 .,9 3,8(. -5 ')43 31 ,-3 39 3'415
E[
ε2min
]
= Var(X)− [Cov(Y,X)]2
Var(Y )= Var(X)(1 − ρ2)6|ρ| ≤ 1
3-8-2-' (43 *+41ρ'),7
.('36-:) ( (),' '+1 *-0 :1 (1 4' .+2 '(8 ) - -21-0;(,3 -',:-23 ')13 5() -/ '(82) 52 * -) 6
(' (:-, *2(2 -21-0;(, -',:-2 ')1 -21 -0;(, -',:-2 ')11 '. (- (' (:-, *2(2 -21-0;(, ')1 66227 3+-+1 ,22 ')11 '. (-,2( (2-,7 ';1 (. (, '1 (27
E[X ], (3
Y3+-+1 * -21-0;(,3 -',:-23 ')13
Cov(Y,X) = 0'),7 66
X2) -',:-2 (')5 '9( (:-,
Y3-82'(4 -'/
Y(X
'),7) ,71 6'5+ ++1:6* --(2. -.25
Y(X
'),7 ,1 (+2 6-21-0;(,3 ')13 ,(3 -21 -0;(,3 -',-:-23 ')13 *-1-(1 *-'415 6'(5E[X |Y ] = c0Y
- (, 3 3'415) 3/7 (3 ,22 -(8 '57 ( '/,1( .,9( - (, 3 3'413 ,-3 .'/, ,1 (+6* -2-'.5 (,5(- . (; (: . (,1 (+ 6*-(1c0
(54-'(04 (3 3'413
E[Y ] = 07 (
E[X ] = 0) /-:: . (0); *)2 6
Y1, Y2, · · ·Yn. (+-+1 ';1 1 2 85.1 (')3 (5 3'412 '(5:) -
Y2) .-',:-2 3-84 :(; , (3) ')12 39 3'415 6
Y1, Y2, . . . , Yn*3 (-5-7') -,'4,3 '(04 (3 ,(3
Y'),7
3,53 3'(83 ., -227 ;(,5X l = aTY =
n∑
i=1
aiYi
,-3 39 3'415 -21-0;(,3 -',:-23 ')13 .,-81 .-5 6*3)27 * - (54 *3aT = (a1, a2, . . . , an)
'),7.('/, *-2-15 6. -21 -:-1 3-3. .8(113 .- (5 -'3 (')3 .,-)) 7
X l = aTY3'(831 ')1 ,81 3-5-(0 -53) 7
(a1, a2, . . . , an)*- (54 ,81
E[ε2] = E[
(
X − X l)2]
= E[
(
X − aTY)2]
6a*- (543 *-'(04 (3 27 -:; 2 -21-:-1 3 -3-
'5-, 5 '(8 -,) 72 .1'( (+-,23 3:.)13 '(5 ( 3+-+13 '(5 ;, 8(11 2) 3/:33) 4 (+5 3'36 (: -0 -:-1'0+ (54 -);(/.8(113 . - (5-'3 (')3 .,-)2 -(0 -53 ., .; (: *; * ()': ('.;E[ε2] = E
[
(X −∑
ajYj)2]
.6
6318)72 35()/ ,-3) 30-)5 .-;(2/ 3/7 (3 -8: )135 6.-1 01 , 3'-) - 3/7 (3 .-: 232*--42 * -7-'8
a∗i*--21-0;(,3
ai-7') 254: .'9:3 ., 2 3 (() :(
ai-;2 2-2 -(0 -53 ., '(9: *,
0 =∂E[ε2]
∂a∗i= −2E
Yi
X −n∑
j=1
a∗jYj
= −2E[XYi] + 2n∑
j=1
a∗jE[YiYj ] 1 ≤ i ≤ n
(')3 .,-) -7 3:(7.3 ., ) - -21 -0;(,3 -',:-23 ')12 -7 *-, (' (:,ε = X −
n∑
j=1
a∗jYj6
6)135 '(9/: (9 3+(4:2 6Yi
.(+-+131 ./, 27 * 2 .5 3-82'(4 .'/
a∗i 2.4
-' (04 ( 5-.75
E[XY1]E[XY2]666E[XYn]
=
E[Y 21 ] E[Y1Y2] . . . E[Y1Yn]
E[Y1Y2] E[Y 22 ] . . . E[Y2Yn]666 666
E[Y1Yn] E[Y1Y2] . . . E[Y 2n ]
a∗1666a∗n
.'8(41 3'(85(E[Y X ] = E[Y Y T ] · a∗6 '5(+1) ((-7 6-+1-1
n'(04 (5
n× n 38-'01 2) 32;71 -:1-3 ,5( -+1-1n'(04 ( -; (1 -2,1)3 ,5 '),7
:, -' ((43 .8-'01) 3/:35 6Cov(Y ,X) = VarY · a∗ 3'(85 * .,9 * ()'2 .-: ;, .2/ (. * *-:.)15- (. : 39 ('.; 67 /, 2;0 : -'2(:-3 3'415
a∗'(5 +-/- , (3 ( ('.; ) - .-'2(:- 3:-,
E[Y Y T ]
a∗ =(
E[Y Y T ])−1
E[Y X ]6
, (3 -21-0;(,3 -',:-23 ')13 (X l
opt = (a∗)TY =
n∑
i=j
a∗jYj6 .'(85
2.18 ., *()'2 ,-3 .'. (:3 3,-)3 ., 5)/2 ./, '+ 6
E[(X − (a∗)T · Y )2] =?.'. (:3 3,-)3 31.-227 3'(85 3-2, '(9/: '), .'/, '+ 6
2.16 (.2 5-832 9, (
C = E[Y ·Y T ](b = E[XY ]
'),7b = C ·a∗
3,-)3 .'+31 3,53 ,-3 )135E[X2] = E
n∑
i=j
a∗jYj + ε
2
6 = E
n∑
i=j
a∗jYj
2
+ E[ε2] + 2
n∑
i=j
a∗jE[Yjε]
.6
72 (:25-4 6E[Yjε] = 0
254: 39 3'415 4'( a∗j
*--2-1 -0;(,3 * -1+413 ., *-'/ (5 '),7 -7 (:-,' *2(,,-3 -',:-2 (' -) '(5 .-21-:-13 .- (5-'3 3,-)3)(E[ε2])min = E[X2]− 2bTa∗ + (a∗)TC a∗ = E[X2]− (a∗)TC a∗
6 (,
(
E[ε2])
min= E[X2]− E
[
((a∗)TY )2]
*-0 :1 (13 '),7 -' (04 (3 3'415 -',:-23 (')3 .-52 3,213 35().3 ., *-:. (:( 2.23)
(( 2.20)
( 2.19)637-;3
C'),7 . (/;2 *-;,.1 (),' '+1
+/, -21-0;(, -',:-2 ')1 4' ) - 37-;3 :, -' ((43 .8-'01) 3'415 -7 3,'1 * 39 / (. -; -7 52 * -) :6-21-0;(,3 ')13 2) . (+-/-2 31 (+5 3-(2. 3,-)3 -7 5(:
( 2.16)3,-)3 ./ (:1 -7 52 * -) : 63,53 ,-3 -21 -0;(,3 ')13 ./ (:2 . -; (2/ 3/7 (3
X, Y*-:.)15 , 13 ., -2/: *, 72 6
EX,EY ,E[YiX ],VarX,VarY5 '1(27 -:) '+ + *-0 :1 (15 4'( ,
7 *, 6-21 -0;(,3 ')13 ./ (: ., * -5(4 '), *-,:.3 27 * 72( 339 ',). 3,-)3 *-0 :1 (1 *. (, -25'), -21-0;(,3 ')13 ./ (: ., * -+(- (:, 39 3'415 6.()15 *-- (, .(-32 (2,7 * -)+/ *-:.)1 '/5:6-21-0;(,3 -',:-23 ')13 ., (()1 4 (-+5 (9 6( 2.5)
3, (()1 -',:-2 , (3,(3 -21-0;(,3 -',:-23 ')13 39 3'415 6
E[Y ] 6= 0, E[X ] 6= 0) 3'415 .7 --:
X lopt = E[X ] +
n∑
i=1
a∗i (Yi − E[Yi])
3-3. .'. (:3 3,-)3( ;, 8(11 *-97'(113 *-:.)13 '(5 2.19
3, (()1 ('.; - *-:(. :a∗i
3 '),7(
E[ε2])
min= E
[
(X − E[X ])2]
− E[
((a∗)T (Y − E[Y ]))2]
632, . (,8(. /7 (3 15 * (17 .-'2(:- 3:-,) :, -' ((4 .8-'01 39 3'415 6,7 * .(;(. -'243 3'415 (:. -:) . ('33., .-1)1 +/ .5 (43 .)'(;1 3,(()1 (:2 ) -+-/- , (3 -21-0;(,3 -',:-23 ')13) '('5 -'243 3'46')13 3:51 ., *-5(4 *3( *-1+413
2.5
E[X ] = E[Y1] = E[Y2] = 0 n = 2
E[Y1Y2] = 0; E[Y 21 ] = E[Y 2
2 ] = 1; E[XY1] = 5; E[XY2] = 7
5
7
=
1 0
0 1
a∗1
a∗2
a∗1 = 5, a∗2 = 7
X lopt = 5Y1 + 7Y2
(E[ε2])min = E[
(X − Y T a∗)2]
= E[X2]− E[
(5Y1 + 7Y2)2]
= E[X2]− 25− 49
E[X2]E[Y 2] ≥ (E[XY ])2
Y1 = X + N1
N1 X 2.6
Y2 = X +N2
N2
1
E[X ] = E[N1] = E[N2] = 0
E[X2] = E[N21 ] = E[N2
2 ] = 1
E[Y1Y2] = E[(X +N1)(X +N2)] = E[X2] = 1
E[Y 21 ] = E[X2] + E[N2
1 ] = 2
E[Y 22 ] = E[X2] + E[N2
2 ] = 2
E[XY1] = E[X(X +N1)] = E[X2] = 1
E[XY2] = E[X(X +N2)] = E[X2] = 1
a∗1
a∗2
=
2 1
1 2
−1
1
1
=
13
13
a∗1 =1
3, a∗2 =
1
3
X lopt =
Y1 + Y2
3
(E[ε2])min = E[(
X − Y Ta∗)2]
= E[X2]− E[(1
3Y1 +
1
3Y2
)2]
= 1− 1
9E[Y1 + Y2]
2 = 1− 1
9(2 + 2 + 2) =
1
3.
X = (Y1 + Y2)/2
E
[
X − Y1 + Y2
2
]2
= 16
3
2.7
E[Y 2
1 ] = 2, E[Y 22 ] = 3
Y1 = X +N1, Y2 = X +N1 +N2 = Y1 +N2
E[XY1] = E[XY2] = 1
a∗1
a∗2
=
2 2
2 3
−1
1
1
=
12
0
6
Y1
Y2 = Y1 +N2 Y2
X = Y1/2 Y1 X
.-'2(:- :, -' ((4 .8-'01*--4 39 3'415 6. -'2(:- ,-3
C = [Cov(Yi, Yj)] :, -' ((43 .8-'01 (5) 3'415 2;02 .7 '(5: /05(17
72 6. (-(2.C
2) . (' ()3 -7C h = 0
) 7 ;,3 '(04 (( (:-,)h
-+1 -1n
(54 '(04 (E[
(hTY )2]
= hTC h = 0
9, (hn = 1
) 7 (545 2-;732 27 (: 72(hn 6= 0
) .(-2273 .25 3 ,22 /-::(h'(04 (5 --:
E[
(Yn + hn−1Yn−1 + · · ·+ h1Y1)2]
= 0
72(Yn = −hn−1Yn−1 − hn−2Yn−2 − · · · − h1Y1-9,
Y1, . . . , Yn−1*- (+- *,5 .('/, *-215 63,-) ,22
Yn., )/:2 27 (: -9,
Y1, . . . , Yn−1*-:(. : *, (:--3+2) .-',:-2 3 -84 :(;7 8--2 ');,
Y1, . . . , Yn2) .-',:-2 3-84 :(; 27 .('-. - ,7 ) - '1 (27 6
Yn.-+-5 '(8 -,
Y1, . . . , Yn(. :)7
X., ')2 * (415 72 6-',:-23 (')3 '(82 +-1 - (1 (:-,
Yn3:.)13
Y1, . . . , Yn−1(. (, 4 (-+5 (5)/3 3-3 - , (3)27i'(5
hi 6= 025,
hn = 0*, 6
Y1, . . . , Yn−1(. :)7
X., ')2 4-;16 (')3 ., 24242 -251
Yi02 '. (2 ');,) 7 +/, . (/;2
i0*--4 -9, . -'2(:-
C*, 27)3 ' (1 6'5+
.-'2(:- ,2 38-'01 ,-3i0
3 3'()3( 3+(13 .4-/1 - .254.1)(n− 1)× (n− 1)
3)+/3 38-'013 *,-23.5 -)1: .-'2(:- --+ 3)+/3 38-'013 *, 6.:04 (13 3-53 '(5 (:-+-5) . (,8(.3 -;2 (')5 -)1:6. -'2 (:- ,2 38-'012 -:) + * (81-8354 .1'(. 3:-, ,-3) ((-7 . (+-+131 ./,1 *2.1 ')13 35
2.731 (+2 31 (+ (:-, 39 3'41) 52 (1 -). (-'2 (:- 2) 3'415 7, ( 6('.;3 2) . (+-/- ' (/ .,051 . (-'2 (:- 6) ' .; (.5 .'/, 3+-+1 3. (-3*, 2)12 6* -39 *3 3) 12 '),7 (,2 *42/(
Yn5 . ()'(;1 * --(2. *42/ *-:() * -7')1 * ()'2 .-:(. (, ., * ()'2 .'/, 3'(8 -3 (9(
X = 2Y3-9,
X = Y1 + Y2 + Y3-21 -0;(, ')1 (:,81 *, (
Y3 = Y1 + Y26-21-0;(, ')1
3-82'(4 .'/ +-1. (')3 .,-) -7 (:-,' -21-0;(, (') '(5 6EX = 0;EY = 0
(5 3'415 97'.: 39 42/5'1 (27 ( 2.13)
3, (()1 .(+-+13 2) 3-84 :(; 27 *
E[
(X − φ0(Y ))g(Y )]
= 0 .6 *-1--41
a∗*- (543 '),7
X lopt = (a∗)TY
,(3Y
++1:)7X
2) -21-0;(,3 -',:-23 ')13) (:,81 7 (17E[Y ·X ] = E[Y · Y T ] · a∗6 '. (- .0'(;1 3'(85 (,
i27 '(5
E[
(X − (a∗)TY ) · Yi
]
= 06
63+-+13 2) .-',:-2 3-84:(; 27 * 3-82'(4 .'/ε = X − (a∗)TY
(')3 .,-) '1 (27+,1 .--1 (97 .-'01 (, .(27.3 6*--'01 (,- *-4 (1 -: '+ *;3 (9 3,8(. .,-812 .; (: '+ 3,': (:,. --1 ,-3 (:5 ( (9 . -((91 (')3 .--5 ., (4.2 '. (- 24 *-.-2 (') . (-5 '(.;2 *-5' * -'4156('.;3 . (:(7. ., -53231 (+ 3:(7. (+1 ( . (+-+13 * 3-82'(4 .'/ 3-3. -21-0;(,3 ')13 2) (')3 .,-) -7 . (;82 ) - (+165/'1 .. 2 -+-24 (, '(04 (( 2) 3203 32-541 3;(. 2 27. : 3:5(. 2542 -+7 -',:-23 3'415 .1--43,-)3 ('(5
a*-1+41 '(04 (( * -);/1 (:, 35
Y, ( 1 2
X, 1 2) -',:-2 (') .--5 (:'+3) -;73,-)3) 7 * -1+41 '(04 (( * -);/1 35 .-'01 (, 3203 .-5 '-+: 7 .-21-:-1
E[(X − aTY )2].- (5 -'363-53 ., .7 0';: *-'(04 (( 2) (, 3
Y.(+-+13 ,7 .-21-:-1 ,-3
X − aTY.- (5 -'3
(. (,5 *--0 -:-1'0+ *-'(04 (Y 1, . . . , Y n
(-3 - 6R
m -+11m
3 -+-24 (,3 5/'15 -0 -:-1'0+ '(04 (X
3-3 -27 ., *-2541θI
*-:.)13 '),7 ∑ni=1 θiY i
- )';:3M(Y 1, . . . , Y n)
5/'13 ..5 --: 6-+11m
5/'12) 372)33 .,X
5 1 : 6(18R
m ,2 6, 69 R
m 2) )11 5/'1 .. 39) /-::( *--');,3 *--)113 *-7'3, (3Y i
(n = m − 1
(5 3'413 2 5()/: 31 (+2 X
2 '. (-5 35 ('43 ,-3)M
5 3+(4 :3 '1 (27M
2X6;, 3(() 3:('/,3 30:-+'(, (43 '(5 . (+(4:3 (, 7 *, 3-3-
M6i
3 30:-+'(, (43 '(,2 '(04 (4'( *, '1 (27 6
(X − X) ⊥ Y i*--4.1
i27 '(5 *, 4'( *,
M2
X2) 20 -33 (, 372)33 , (3
X3:06. (+-+12 .58: (')3 .,-) *,
-0432 .-: -9, . (58-: -, *, -7 52 *-) ( 3,-)3 5() -/2 '(.-; 0;)1 ., 2;3 3:(7: 3:03 (+1 -532 -+763,-)3 .,-+- 2 )';:3 5/'13 .. 2
X2)
X372)33 ,81 6
Rm5 *-'(04 (
X,Y 1, . . . , Y n*-:(. :
Rm 5 32,)6Y 1, · · · , Y n
Y i
*-'(04 (3 2) -',:-2 ('8 , (3X
35().X = h∗1Y 1 + h∗2Y 2 + · · ·+ h∗nY n
i27 '(5) 5(4 372)33 ('4 (
(
(X − X), Y i
)
= 0.(-',:-2 . (, (()1n2) (,3 - '+ (1
h∗72
(X,Y i) =
n∑
j=1
h∗j (Y i, Y j)
37237 '-+:) -,:.5 *--+11 ( -, *-5/'12 35/'32 .-: 39 25,R
m -+11m
5/'15 (:/ -: 3,8(.3 .,'. (- 4 --(+1 ;(,5 632, *-5/'15 .-'24 32;71 2) ) (1 (:. ()'2 3-3-( 32,7 *-5/'1
x, y
(x, y) 2.8
x
S ||x− y||2 = (x− y, x− y)
ε .= x− x
x S x S
z ∈ S
(ε, z) = 0 S
(−1)5 2;7 ( ||z|| 5 34 (2/ -+- 2 6
(ε, z) = α 6= 0) 7
S5z*--4 '1 (27 3:(7: 3:-, 3:03) /-:: 35('4 )11 ,-3) /-7 (:(
S5x3)+/ 3+(4: .7 '-+: 6
(z, z) = 17 (
α > 0-7 /-:32 ');, '(83 .+-15
((-7 ( -',:-2 5/'1 , (3S
) ((-7 -9, 6x = x + αz
'-+: 63 :03 ., /-7 (: 75 ( 3'-. 3-3. .,9 x2 '. (-2) . (-',:-231 -7
z2 .7:(,1
x'(5 3,-)3 -7 52 * -) : 6
S5 , (3 * -',:-23 ('-83
S5 *3
z* (
x* ).-'243 32;713
(x− x, z) = (x− x− αz, z) = (ε, z)− α(z, z) = α− α = 0.6
x
(x
-5 4/'13 ., 5)/: .7||x− x||2 = (x− x, x− x)6
= (x− x− αz, x− x− αz)6 = (x− x, x− x) + α2(z, z)− 2α(x− x, z)6 = ||x− x||2 + α2 − 2α26 = ||x− x||2 − α26
.1--4) -,:.5 5 (17 .(58-: . :(7. *--42 5--/ -21 -0;(, ')1 27 -7 (:/7 (3 75 6x
2 '. (- 5 ('4x
7, (6.-1-:; 32;714 (+5: 6. -21-:-1 .- (5-' 3,-) * 20 -33 .-5 -52 , 1 2) -',:-23 (')3 .-5 -5 325435 '79-: *(7-2
3,53 3-(2:,3 .,R
m 5 * --0 :-1'0+ * -'(04 (Y 1, Y 2 ←→ E[Yi] = 0
*--'24 , 1Y1, Y2
(Y 1, Y 2).-'24 32;71 ←→ Cov(Y1, Y2)
‖Y ‖ = (Y , Y )12 =
'(04 ( '(, ←→√
Var(Y )
(Y 1, Y 2) = 0.(58-: ←→ Cov(Y1, Y2) = 0
Y =
n∑
i=1
θiY i*-'(04 (5/'1 .. ←→ Y =
n∑
i=1
θiYi, 1 -',:-23 3'415 5/'1 ..
←→ Y = g(Y ) g27 , 1 -21-0;(,3 3'415 5/'1 ..
65/'13 ..5 '(04 (( 272 .58-: 3,-)3) 7 -+- 2 .:--; (,1 32033 3-(2:,3 (272 254:(
Rm5 3,8(.2 3-(2:,3 .250 ., 2-;:
Y1, . . . , Yn (.1
X, 13 2) -',:-23 (')3 .-52 '(5:
i
Cov(Yi, X) =
n∑
j=1
h∗j Cov(Yi, Yj)
'(5 .(58:3 ('4 ., 254: 339 3'(85 6. (:(7 : . (,8(. * -:. (: 372)33 ('4 ( 3 -(2:,3 72( 2.28
4 (-+5 39(63,-)3 .(58-: -+- 2 --;(,1 -21-0;(,3 )13 -21-0;(, (')*-7-2)1 (:, (-2) 5/'13 ..2 .58-:
X − X = X −∑ a∗i Yi-21-0;(,3 -',:-23 ')13 2) 3,-)3 *7 :
(:--3+E
[(
X −n∑
i=1
a∗i Yi
)
· Yj
]
= 0; j272
* --4.1) .(58:3 225 *-+(- (:,R
m5 .'. (:3 3,-)3 -52‖X‖2 = ‖X‖2 + ‖X − X‖2
(')3 .-5 '(5 )'(;1 5 () -/5 .,9 3,':E[ε2] = E
[
(X − X)2]
= E[
X2]
+ E[
X2]
− 2E[
XX]
= E[
X2]
+ E[
X2]
− 2E[
X2]
= E[X2]− E[X2]
-',:-23 (')3 .-5 2 5 ()/2 .-: * (7-2 6E[
XX]
= E[
X2] .(58:31 5(: ('/, -:;23 (-()3 '),727 -+- 2 '1 (27
Yi3 -+- 2 )';:3 5/'13 2
X2) 20 -3 . -5 27
Y1, . . . , Yn, 1 (, (.1
X, 1 2)
.:--;,1 (9 3 :(7. ( 3+-+1 272 .58-: 3,-)3 ('(5 +-/-3 , (3 -21-0;(,3 ')13 6*32) *--',:23 *-; ('-836-21 -0;(,3 -',:-23 ')13 .,(2-, ( -21-0;(,3 -',:-23 ')13 ., --;,1
E[(X −∑ a∗i Yi) · Yj ] = 0, j = 1, . . . , N52 * -)
('4 / :2 72 ');, 6.-:. (13 .2/ (.3 -21-0;(,3 ')13 ., --;,1E[(X − X(Y )) · g(Y1, . . . , Yn)] = 06-21-0;(,3 (')3 '(5 * 372)3
-,'4, 3:.)1 3 ) (1 2) 35/'3 -3 (9 *--,'4, *-:.)1 2) -;( (, , (3 8.14
3'+3 -,'4, '(04 ( (:5 6-; ( -, . (-32 2(7 - * -:.)13 ';1 '),7 .; (: 35/'3 , (3 -,'4, -23. ) (13 6* -+1-1 ';126. (54 ( 19 . (+(4 :5 (-; (3 (2-,7 *-:.)13 (,2 *-/--.1 (:,. (,2) /-:: 6 3';-3 (, 3';-3 .'+()1
k19 .+-/- 275 '1 (27 -.';- .(, * -'+)1) /-::
3.1 .(,3 6n(k)
) '1 (X(k)
'+()13 2+(31 57'(13 X(k) + n(k)
., *-02(4 (:,k
195 '1 (27 ) ' ((.1 396-,'4, -23. *-,'(4 397 . (,2 619 .+-/- 275 +/, 3:.)1 *--,'4, *-:.)11 57'(1 72 * -02(4 (:,)3:.)1 2) .-,'4, 3-84 :(; 27 * (-2 5()/2 ');, 6, 1 2) 3'+- ,(3 +-+5 195 -,'4, -23. '1 (2719 3:.)1 2) .-,'4, 3-84 :(; , (3 '), -8' 195 -23.2 35/'32 .:.-: (97 . (27.3 6193 3:.)1 +-+56 -8'2541
t'1 (27 X(t), T1 ≤ t ≤ T2 *--,'4, *-:.)1 2) 3'+- , (3 +-+5 195 -,'4, -23.
3.2 6 (54ω = ω0
'(5t
193 3:.)1 2) 3-84 :(; 27 ,-3 -,'4,3 -23.3 2) * +1 . --84 :(; 6*-+-+5 *-7'3'(831 3-84 :(; 27 -3 (9 '1 (27X(t, ω0), T1 ≤ t ≤ T2,
(54ω0
,'4: +-+5 195 -23. 6(−∞)
(,0'3 ., 254 -
T1(
+∞ (, -;( 3-3-T2
*-12) *-7' 2541t227 '+5
(-82 (2, . (-. (,5 *-.-2 )1.): 72( i, j, k, l,m, n
. (-. (,5 *-12) 12 25 (41 6. () 21 .-. 3'+ * *-.-2 *2(, Xn(ω)
(,X(n, ω)
-+- 2 +-+5 195 , . 1 : 227 '+5 6+-+5 195 -,'4, -23. 2) 193 3:.)16 (7 (Xt
(,X(t)
*-2(4)3 *-'8(413 *-:(1-3 -:)5 )1.): . (5('4
(:, -,'4, -23.2 (9 3'+3 5-/'32 (:, (55 *2(, 68.15
3'+3 (2-; .-84:(; '-+32 .-: -,'4, '(04 ( ' (5.(-('5.3 5)/: +8-7 '('5 ,2 19 . (+ (4: ( -, '(5 '+(1 . (-32 2(7 - -,'4, -23.) ((7 *--)45 *-24.:6,53 2-'.5 .',(.1 . (-53 ./, 6* -:.)1 ( -, -+- 2 *-'+(13 .('(,1 2)',.1 .(-(2. -.25 501 . (4 -'9 * -',.13 Xi(ω), i = 1, 2, . . . *--,'4, *-:.)1 .'+ 3:(. :
3.3 -; ( -,3 (2-; 3 ., 5)/2 +: -7 3,'3 6-2; ',.1 &
P X1 ≤ a1, X2 ≤ a2, . . .(3.1)
t1 <*-4+:-, *-:19 .'+- 272
N272 6,53 -,:.3 *--4.1 *, 4'( *,
a1, a2, . . ..-;( -, 3'+- 272'(04 ((3 2) (2-;3 ., 5)/2 * -+(- (:,
N'(,5
a1, a2, . . . , aN3'+- 272(
t2 < · · · < tN
FX(t1),X(t2),... ,X(tN )(a1, a2, . . . , aN ) = P X(t1) ≤ a1, X(t2) ≤ a2, . . . , X(tN ) ≤ aN
.,( 0,-3
( 3.1)5 . ('5. -33 9, ( -; ( -, , (3
ai < 1) *-1;3 ';1 (5 3'413 ., +';-:5 4 (+5 91'6, 1 2) -;( (, -+- 2
( 3.1)., * ()'2 .-: 9, ( *-1; 2) -; ( ';12 0';
ai ≥ 1(5 3'413
'+5 *-,:. 2) -; ( -, ';1 - '+(13 '(,1 2) . ('5.3 5)/2 (:, (55 -7 , (3 39 3'415 -'4 -3 -) (4363,53 322735 '932 / (: 72 6 3(() 3--3. (9 . ('5.3 227. ('+(13 . (;. ()13 (2-;3 . (-84 :(; 27 (, , (3 +-+5 195 -,'4, -23. 2) . ('5.33 4 (/
3.4 -; ( +1-15 * - (2-;3 27 2) (,3 (3 9 '1 (27 6. -; ( *-:19 .'+- 27 '(5FX(t1),X(t2),... ,X(tN )(a1, a2, . . . , aN ) = P X(t1) ≤ a1, X(t2) ≤ a2, . . . , X(tN ) ≤ aN, (3)
FX(t)2+(3 6
a1, a2, . . . , aN3'+- 27 (
t1 < t2 < · · · < tN*-4+:-, *-:19 .'+- 27
N27 '(56
t195 -23.3 2) -2()3 (2-;3 (, -+1-1 +/3 (2-;3 ,'4:
X(t)-,'4,3 3:.)13 2) (2-;3
* --5 (-/ *-:1 9 ' (5 '+(13 -23.5 :(5.: 31 (+2 6-23. 2) (2-; ',.1 *--; ( *- (2-; (, 27 ,2) 5 (17* -:(-(()3 -:) * -1 --4.1) 7.- *,3 6X(1), X(2), . . .
P X(1) ≤ a = 0.5
P x(1) ≤ a, X(2) ≤ b = 1
.(-32 5--/ -7 ' ('5 72 6-:) 2 '),1 '. (- 32(+ . ('5.3 -,+((5 (),'3 '(,12 7) 7.- ,2 -,+( 39 .-:1 9 (56, (3)27 . (-54- -, :.63,53 . (-0 :0 - :(4 .(-54-3 .) -'+ ., *--41 , (3 *, 4'( *, -23. 2) (2-; ',.1 *- (2-; (,
272(t1 < t2 < · · · < tN
*-4+:-, *-:1 9 .'+- 272 1 ≤ k ≤ N
(N ≥ 2
272 .(-54- .) -'+3.5 -,:.3 *--4.1
a1, a2, . . . , aN3'+-
P X(t1) ≤ a1, . . . , X(tk−1) ≤ ak−1, X(tk+1) ≤ ak+1, . . . , X(tN) ≤ aN
= P X(t1) ≤ a1, . . . , X(tk−1) ≤ ak−1, X(tk) <∞, X(tk+1) ≤ ak+1, . . . , X(tN ) ≤ aN
(2-; . (-84 :(; 2) (1 -5 (,FX(t1),... ,X(tk−1),X(tk+1),... ,X(tN )(a1, . . . , ak−1, ak+1, . . . , aN )
= FX(t1),... ,X(tN )(a1, . . . , ak−1,∞, ak+1, . . . , aN )
6'. (- * -7 (1 : *-+115 (2-;3 .-84 :(;2 * -,.1 *-3 (5 *-+111 (2-;3 .-84 :(; 2) -2()3 (2-;3 '1 (276T1, T1 + 1, . . . , T1 +N
3'(831 19 . ('+5 )1.)32 4 -;1 3.4
3'+35 -7 3,'33.6
6+-+5 195 *--,'4, . (. (, 2) . (,1 (+ ';1i.i.d
3'845 (2-; -(() ( . -0 -00 *--(2. -.25 * -:.)1 2) (, X(n), n = 1, 2, . . . 3-3-3.7 '),7 +-+5 195 -,'4, -23. 2,7 '+(13 (,2 /--.32 ');,
= independent, identically distributed52 ) ' ,'4: -23.3 -9, * -- (, *3 , 13 *, 652 ) ' ,'4: 397 -23. 6
t = 1, 2, . . .*-12) *3 *-:1936- (, 52 ) '2 * -:((7.1( 52 ) ' /:(15 * -)1.)1 *-.-2 -7 52 * -)2 ) - 6- (,
2+(1 *-5' *-'415 (:5 6* -' (1-3 .:(71 2) . ('9(/ . (2;35 . (-793 .'+-7 2)12 -; (1 52 ) ' -23.6-24 -9-; )'2 2+(17 - (, 52 ) '5 * -)1.)1 *-5' * -'415 0';5 6397 , (3 . (+-+15 ) '3FX
5 1 : 63 9 (2-; 4 (/ 5)/:( 523 )'3 -23. 2) (2-;3 4 (/ ., 5(4X(1)
, 13 2) (2-;3 -7 3,': 353ai, i = 1, 2, . . . (-3 - 6*-'/,3 *-:.)131 +/, 27 2) (2-;3 4 (/ * , (3)
X(1), 13 2) (2-;3 4 (/ .,254: t1 < t2 < . . . < tk
*, .(2.3 -, 225 -9, 6*3)27 *--)11 *-';1FX(t1),X(t2),... ,X(tk)(a1, a2, . . . , ak) = FX(t1)(a1) · FX(t2)(a2) · · ·FX(tk)(ak) =
k∏
i=1
FX(ai)
-,'4, -23. '-+: 6 (2-; -(() ( .5 *-)';3 -23. , (3 '(7-) (2-3 (, -,'4, (2-33.8 -+- 2 Y (n), n = 0, 1, . . .
Y (0) = 0, Y (n) = Y (n− 1) +X(n) n > 0
.'/, '1,- ,2 *, 6 .5 *-)';3 -23. ,'4 -Y (n)
-23.3 .5 *3 X(n), n = 1, 2, . . . *-:.)13 '),76 (2-; -(() *3 X(n), n = 1, 2, . . . *-:.)13) /-:: (:,* -101
drunkard walk (, random walk
* ,'4: (2-; -(() ( .5 *-)';3 -23.*-)';33 '+ 397 -23. '-+32 25(41( ');, 6*-:5 (1
Y (n) =
n∑
i=1
X(i)(3.2)
6291 .:(715 * -'(1 -3 .'+-5 '50813 /(('3 , (3 397 -23.2 . (-) 1 . (,1 (+* -:(1-5 6. ('5.33 .-84 :(; ' (5 0 (); -(0 -5 * ()'2 .-: *-12) *-7' 2)12 2541 ,(3 *, 397 -23. '(5
3.7
31 (+ 2)P Y (1) = a1, . . . , Y (k) = ak
= P Y (1) = a1, . . . , Y (k − 1) = ak−1, Y (k − 1) +X(k) = ak
= P Y (1) = a1, . . . , Y (k − 1) = ak−1, X(k) = ak − ak−1
= P Y (1) = a1, . . . , Y (k − 1) = ak−1 · P X(k) = ak − ak−1
/.;:( -)1: 31 (+ 3'(85 6. (2.3 -, 225P Y (1) = a1, . . . , Y (k) = ak
= P X(1) = a1, X(2) = a2 − a1, . . . , X(k) = ak − ak−1
= P X(1) = a1 · P X(2) = a2 − a1 · · ·P X(k − 1) = ak−1 − ak−2 · P X(k) = ak − ak−1
=
k∏
n=1
P X(n) = an − an−1
254: * -1 (+ * -2(4 -)1 -9, . (;-;8 -25 * -:.)1 *3 X(n) *, 6a0 = 0
*-'-+1 ('/,3 '513 '(82 '),7fY (a) = fY0(a0) ·
n∏
i=1
fXi(ai − ai−1)
6
A, φ X(n) = A cos(ω0n+ φ)
3.9
a1, . . . , an X(n)
A cos(ω0k+ φ) ≤ ak k P A cos(ω0k + φ) ≤ ak, 1 ≤ k ≤ n
A, φ
(X(t1), . . . , X(tN ))
N, t1, . . . , tN X(n) 3.10
6 (+1 - (, , . ,(3 - (, ,(3 * --,'4,3 *-:.)131 +/, 27 2) (2-;3 (' (5 52 ) ' 2)12
'(5 63 -82' (4 (0 (, . -84 :(; ( :, -' ((3 .2/(.3 . --84 :(; *3 -,'4, -23. *-',.13 '. (-5 * -- -53 *-2+3';1 *3
8.233'+3 :, -' ((43 (
8.223'+3 .(:()3 *-0 :1 (13
8.173'+3 .2/(.3 *--,'4, *-:.)1. (::(5.3 - -:) '+ 22 (7 + *-0 :1 (1 . (22(73 -,'4, -23. 2) . (2-541 . (:(7. '-+32 27 (: *.'95 6* -6
t1, t2*-:1 95 * --(2. 5 (17 (-3- * -2+3 6
(X(t1), X(t2))*-:.)13 (95 (,
X(t1)-,'4,3 3:.)15
., '-+: *2)t'),7 X(t), −∞ < t <∞ -,'4, -23. '(5
3.11
µX(t) = E[X(t)].2/(.3 .--84 :(; 6
RX(t1, t2) = E [X(t1) ·X(t2)]3-82'(4 (0 (,3 . -84 :(; 6
:, -' ((43 . -84 :(; 6KX(t1, t2) = E [(X(t1)− µX(t1)) · (X(t2)− µX(t2))]
= RX(t1, t2)− µX(t1) · µX(t2)
'(9-;3 ( -0 -:-1'0+ 5('-4 8(113 4' ,2 -23.3 2 .( .(7'3 . (4;1 3-82'(4 (0 (,3 ( .2/ (.3 . (-84:(;6. (:() 19 . (+ (4:5 -23.3 -7' 2) .-',:-23 .-0 -003 . (2.3 2) 37'3 * ,2, :, -' (( 63.8
31 (+ '(7-) (2-3 ( 3.7
31 (+ 52 )' '(5 (2, . (-84 :(; 5)/: 353X(n)
-23.3 '(53.12
µX(n) = E[X(n)]
= E[X(1)],-3 3-82'(4 (0 (,3 .-84:(; 6n
195 -(2. (:-,µX(n) = µX
72( n5 -(2. (:-, (2-;3) ((-7
RX(t1, t2) = E[X(t1)X(t2)]
=
E[(X(1))2] t1 = t2*,
(µX)2 t1 6= t2*, 6
) ((-7 6* -:() *3 *-:193 '),7X(t2)
-52X(t1)
-5 .-0 -003 . (2.3 -, 225( 195 . (2.3 -, 225E[(X(1))2] = E[((X(1)− µx) + µx)2] = σ2
x + 2 E[µx(X(1)− µx)] + µ2x = σ2
x + µ2x
6 .-2273 3/ (:3 ., (:25-4
RX(t1, t2) = µ2x + σ2
xδ(t1 − t2) . 6
-7 +--1 5 (: ,71 6.227 (13 3-84 :(;5 ,2(+-+5 195 '4:('4 2)δ3 .-84 :(;5 '5 (+1
KX(t1, t2) = σ2xδ(t1 − t2) .
6 6t1 − t2 )';33 '+ 4' 193 -:.)15 *--(2.
RX(t1, t2)(
KX(t1, t2)-7 52 * -) :
8.20
3:0 .2/(.3 2) . (-',:-23 .'95 254:(( 3.2)
(8--5 '9: Y (n)
-,'4, (2-3 '(5µY (n) = E[Y (n)]
=
[
n∑
i=1
E[X(i)]
]
=
n∑
i=1
µX(i)
30 ();3 3/ (:3 ., 254: (2-; -(() *3 *-)';33) ((7µY (n) = nµX
k ≤ n '(5 .7
RY (k, n) = E[Y (n) · Y (k)]
= E
[
n∑
i=1
X(i)
]
·
k∑
j=1
X(j)
= E
k∑
j=1
n∑
i=1
X(i)X(j)
= E
k∑
j=1
n∑
i=1, i6=j
X(i)X(j) +
k∑
i=1
(
X(i))2
'57 (:5) -/ 6i 6= j
'(5E[X(i)X(j)] = µ2) 7 .(2.3 -,5 )1.) :(
σ2 = E[(X(i)− µ)2](µ = E[X(1)]
1 :254 : i = j
) 7i ≤ n 2) +/, ' 4(-+5 (:) -
j ≤ k 272) ((71 (E[(X(i))2] = σ2 + µ2)
E[Y (n) · Y (k)] =k(n− 1)µ2 + k(σ2 + µ2)
=knµ2 + kσ2
=RY (k, n)
7 (KY (k, n) = E[Y (n) · Y (k)]− E[Y (n)] E[Y (k)]
6 =knµ2 + kσ2 − (nµ)(kµ)
6 =kσ2 .
6 '(7-) (2-3 '(5 *(7-2
RY (k, n) = kσ2 + nkµ2 k ≤ n '),7 6
KY (k, n) = kσ2 k ≤ n '),7 6
6-23.3 2) (2-;3 ., *-5(4RY (k, n)
(µY (n)
-7 5 (: . ('+331 -9, - (, , . ,(3 Y (n) *, -7 52 * -) :63,53 3'(85 (2-,7 * -1.-' (2, * ()'2 25(41 6* --5-'(4' *-1.-'(2, *-:/(5 *, .254.1 '. (- . -227 31 (+
-3. 6.-0 -00 *--(2. -.25 , 11 57'(13 -,'4, -23. 52 ) ' X(n), n = 0, 1, . . . 3-3 - 3.13 - Y (n), n = 0, 1, . . . -23. '-+: 6Y (0) = y0
., 54:( 3:(. : 3-84:(;g
Y (n+ 1) = Y (n) + g(Y (n), X(n)), n = 0, 1, . . . , Y (0) = y0
*2(, 63. (, /.:2 27 (:) -+71 -+1 .-227 ,-3 (9 3/ (: 6)' . (/7 (:5 -5-'(4' *.-'(2, 2) -227 / ( -: (392)12 *-0 (); *-'4 -1 /.:2 .-:
(2-; -(() ( .5 , 1 2) (, X(n), n = 1, 2, . . . -3 - 3.14
P X(1) = 1 = P X(1) = −1 =1
2
'(5 . (,/ (: * ()' (, ., 5)/ 3.13
(3.8
3.7
.(,1 (+ ' (5 6g(y, x) = x − y
2
3-84 :(; '-+:(Y (0) = 0
3-3 -MATLAB
.'95 -(8' 00') 6RY (t1, t2)
(RX(t1, t2)
3-82'(4 (0 (,3 . (-84 :(; ( µY (t)
(µX
.2/(.3 . (-84:(;* +13 . (-84:(; *, * +13 . (-84:(; 2) 8(113 ., ( -23.3 2) * +1 . (-84 :(; ';1 .2/(.3 . (-84:(; .,,(3 * +13 . (-84 :(; 2) 8(113 -9,X(t, ω1), X(t, ω2), . . . , X(t, ωk)
31
k
k∑
n=1
X(t, ωn)
.(-84 :(;3 ., 7 (RY (1, t)
(RX(1, t)
3-82'(4 (0 (,3 . (-84:(; ., 00') 6t
193 3:.)1 2) 3-84 :(; ,-3)CX(1, t;ω) = X(1, ω) ·X(t, ω)6. (:41 43 6(2-, . (-84 :(; 2) 8(113 ., (
ω2) *-7' ';1 '(5
Y (n) 3.15
Y (n+ 1) = aY (n) +X(n) , n ≥ 0, Y (0) = 0 6
Y (n) |a| < 1
µy(n+ 1).= EY (n+ 1)
6 = E[aY (n) +X(n)]
6 = aµy(n) + µx(n) .
6 µy(n)
µx(n) = µx µx(n)
µy(n)
µy = aµy + µx 6
µy = µx/(1− a)
n
µx(n) = 0
σ2y(n) = EY 2(n)
6 = E [aY (n− 1) +X(n)]
2 6 =a2σ2
y(n− 1) + 2aE[Y (n− 1)X(n)] + σ2x(n) .
6
X(n), Y (n − 1) X(k), k ≤ n − 1
Y (n − 1)
E[Y (n− 1)X(n)] = 0
σ2y(n) = a2σ2
y(n− 1) + σ2x(n) .
6 X(n)
a2
σ2y(n)→ σ2
x
1− a2.
6
6AR: Auto-Regressive
-23. 2) -0'; 3'41 , (3 (9 31 (+5 ', (.13 -23.3, . 2)12 *--' (04 ( * -7-23.2 .-+--1 3'(85 5-/'32 .-: * --,'4, *-7-23.5 *-4 (3 39 4';5 * -) (13 .,5-/': . ('+33 ',) ., ( *-' (04 (3 '+ '+(1 . ('5.33 4 (/ 6* --,'4, *-'(04 ( 2) (, , (3 X(n) -'(04 (6*,.35'01';3 '),7
v = i+a3, (()13 ., *--41 (2) 3+(53 * (/.5 '), -',:-2 -:('042, 5-7' (:-+-5
3.16 n3 3+-+13 7) 4-(+15 +(+12 * -2(7 - (::,
v., 6 (+-( (54 ' 2
i., 54: 6 (+- (:-, (7' , (54 , (3
a
X(n), n = 1, 2, . . . ) '3 -7 * -+(- (:, 6) ' +( ( 3+-+1 '1(27 v(n) = v +X(n) = i+ a+X(n)
., .:. (:a2) '3 ) (;-/2 ,53 *.-'(2,3 ., *-2-;1 (:, 6
08(11 * (2-; -(() ( .5 *-:.)11 57'(1
a(n+ 1) = a(n) +1
n[v(n)− (i+ a(n))]
= a(n) +1
n[(i+ a+X(n))− (i+ a(n))]
6*-++1: (, * - (+- * -2+5 4' )1.)1 , (3 (8-52 .-: *.-' (2,3 -7 52 *).(3:.3 ., ',. (
b(n)-23.3 '(5 .)'(;1 3/ (: * ()' 6
b(n) = a(n) − a 35833 -+- 2 39 *.-'(2, 0);62(+n 5' . (-8'0 -, ';1 '/,2 *.-'(2,3
54
.57'(13 35(. *-2541 35-8-( 195 3 (54 .-',:-2 .7'1 '+ -' (9/1 . (, (, 195 (54 . (, * -'-51 '),73;(. . (,'2 ');, 63 -:73 . (,2 *.1,.3( 32/.33 -,:.2 3'()4 '51 .;(. 65 -8- 5811( '51 .;(.1666--:1 (:-, , (3 , (54 . (,2 2-5413 , (3 -,'4, 2+(2 0'; -2(, (54 -23. 6* --,'4, *-7-23.5 31 (+(2, . (. (,2 6* --':(-803 . (. (,3 *3 195 . ( -54 2) '. (- 3)2/ 3:(7. -25 . (. (, 2) '. (- .:--:1 3/;)1.7'15 '5 ( 397 . (, '),7 *,3 2(,)2 27 (:( 232 '-+:) 5 (15 195 3 (54 ,-3) .-0 -00 .(3:.3 ) -6195 . ( (54 . (-0 -00 .(:(7. 25 ,(3 ,8(13 . (, * 195 3 (54 ( . -',:-2272(
τ (54 272 *, -':(-80 ,'4: X(n), −∞ < n <∞ +-+5 195 -,'4, -23. 6. (-':(-80
3.17 6X(t1 + τ), X(t1 + 1 + τ), . . . , X(t2 + τ) 2) (2-;2 339 X(t1), X(t1 + 1), . . . , X(t2) 2) (2-;3t1 < t2619 . (993 ./. 3:.)1 (:-, (2-;3 '1 (276
τ > 027 (
t1 ≥ T 27 '(5 *--4.1 -,:.3 *,T
191 2/3 -':(-80 -23.3) '1,:1, 2, 3, 4, 5, 6 *-7'3 ., 2541
Z'1 (27 3-5(4 .4-'9 ',.13 , 1
Z3-3-( , (3)27 , 1
Y3-3-
3.18 63 (() . ('5.356-:1 ,2 -7 *, -':(-80 -23. 5(17 (39 6
n272
X(n) = Y'-+: 6
-23.3 -9, 63 (() . ('5.35 1,−1 *-7'3 ., 25413 -':-5 , 1B
3-3- 6B(n)
.= B · (−1)n 6
3:.) - ,2 ,(3 -9, -(9 ';1 , (3)τ
5 -23.3 ., 9-9: *, -7 52 * -) : .,9 . (,'2 -+7 6-':(-80 , . ,(34 (-+5 2(;1(−B)
*2(, 6(−B)
5B
., (:;2/3 (2,7 3,': )+/3 -23.3 -(9 -, , (3τ*, .,9 .1(2 62276
B(17
*2) (:-,α*, 6-':(-80 -23.3( (),'3 3'415 (:, -9, *2)
α*, 6
X(n) = Y [1 + sin(παn)].7 '-+: 66
FX(n)-+1-1 +/3 (2-;5 :(5.32 4-;1 .,9 . (,'2 -+7 -':(-80 (:-, -23.3 -9,
, (3)27τ'/5: .,9 . (,'2 -+7 6-':(-80 , (3 -23.3 -9,
α = 1/3*, 6
X(n) = sin[απ(n + Z − 1)]'-+: 6-9, 6*2)
k'(5
m = [m] mod 6 + 6k*--413
52
0-5 *2) , (3
[m] mod 6 Z = [Z + τ ] mod 6 + 1
'-+:(581 6X(n)
-23.3 2) (2-;2 339X(n) = sin[(π/3)(n + Z − 1)]
2) (2-;3 72( 3-5 (4 .4-'9 ',.1Z
* *--4.1 62) '(,5 '(9/1 * .-'(9/1 3-84:(; , -3
sin[(π/3)n]) ((71 -:)
sin[(π/3)(n+ Z − 1 + τ)] = sin[(π/3)(n+ [Z + τ − 1] mod 6)] 6
= sin[(π/3)(n+ Z − 1)] 6
6-':(-80 ,(3 -23.3 X
2) (2-;2 339X
2) (2-;3) ((-7 6X(n) = sin[(π/3)(n+ Z − 1 + τ)]
72(3,8(.3 6-':(-80 (:-, -23.3 α
2) *-'/, *-7'2 -227 ;(,5( α2) '5 3-(2. 3,8(.3) 52 * -)3'(85 2 (;13 -,'4, 3:.)1 ,-3 (2) 39,;3 '), -'(9/1 . (, ,53 ('43 2) -0'; 3'41 ,-3 (:25-4)62-2 .,92 339 3/7 (33 6-':(-80 -23. , (3 *2) '(9/1 2 3+-/,
254: *-7' (2-,2 *-'/, *-7' '(5 /'735 ,2 , α = 1/3
'(5 -':(-80 , (3Y sin(απn + Z)
-23.3 6. (-':(-80
254: 6X(n)
3:.)13 2) (2-;3 .,Fx
5 1 : 52 )' '(5 6 P X(t1) ≤ a0, X(t1 + 1) ≤ a1, . . . , X(t2) ≤ at2−t1 =
t2−t1∏
i=0
Fx(ai) 6
619 .993 ./. 3:.)1 (:-, (2-;3) '('5((:-, 39 -23. 72 6195 -(2.
Y (n)2) (2-;3 (2-;, 72( 195 -(2. :, -' ((3 -7 (:-,' '(7 -) (2-3 '(5 66-':(-80
(:, *, 6(:+-5 -(81 *-(2-;3 2 +-13 27 '),7 * 4 (+52 3)4 '), 3:(7. ,-3 . (-':(-80 -7 -+12 '('5'), .(-':(-80 2) '. (- )2/ ( '-+32 .-: -9, -:) ( (),' *-0 :1 (1 -+- 2 -23.3 '(,.5 4;.32 *-:7 (1*-1 () -- 5 ('5 34-;1
wide sense stationary X(n), −∞ < n <∞
3.19
EX(n) = µx(n) = µx
n2 − n1
K(n1, n2) R(n1, n2)
6'83 5 (15 . (-':(-80 -' (413 5(15 . (-':(-802 ,'4: * -2+533 ., ++/2 '(8 ) - '),7'. (- 324 34-+53) * '('5 65/'3 5(15 -':(-80 * , (3 -':(-80 .(, 27) 5(17
3.1831 (+2 /--.35-':(-80 (:-, '(7 -) (2-3 .,9 .1(2 6* -,:.3 ., *-1--41 *3( 52 ) ' '(5 -:) ( (),' 0 :1 (1 (:5) -/ 2)126195 -(2. -:)3 0:1 (13 ;, 8(11 2) 3'415 * 195 -(2. 8(113) ((-7 5/'3 5(15
.-(2 (:70 (, .-5-0 3;(. ',.2 * - :1 (:, '),7 6(:2 * - (+- * -7-23.3 (, , 13 2) *-(2-;3) (:/:3 37 +*, .(/:33 ., 4(+5: +8-7 *2(, .(/:3 /-:32 5(17 ');, 6397) +- (:2 -, *--,'4, *-:.)1 2) 2+(1 -(:-, ( 503 - '/5:
ω -23.3 2) ./, * +1 . --84 :(; 4' +(+12 27 (:) -'3 852 (:-+-5) . (+-+13 2 5()/:6(:+-5
2 71 +(122( 4-;1 (', 19 '(,2 ./, * +1 .--84:(;5 :(5.32 .-: *'(5 *-7-23. 2) 3/;)1 )- ,6* --+(',3 *-7-23.3 ./;)1 -3 (9 6-23.3 2) *-(2-;3* --4.1 *, -+(', ,'4: X(n), n = . . . ,−2,−1, 0, 1, 2, . . . -':(-80 -,'4, -23. 6. (-+(',
3.20
*-:.)1k + 1
2)g31(/ 3-84 :(; 272(
k272
limN→∞
1
N
N∑
n=1
g(Xn, . . . , Xn+k) = Eg(X0, . . . , Xk) ∀ω(3.26)
6 .('5.3 3--3. (-(() ) -) '(,12) '1 (27 1. ('5.35 *--4.- (-(()3) ) ('+: 4 (-+3 12
.:--81 3-84 :(; '-+32(a';1 '(/52 2)12 27 (: -+(', -23. (:-+-5 *,
1(−∞,a](x) =
1x ≤ a *,
0.'/,
) ((-7E[1(−∞,a](X(0))] = 1 · P X(0) ≤ a = FX(0)(a)
',3 8(113 5() -/ ( 19 '(,2 -23.3 2) 3+-+1 - *--,'4,3 *-:.)13 2) (2-;3 ., 5'42 '+ (:25-4 (+- (:-,) -0 -003 ,2 -01.-
FX(0)(a) = limN→∞
1
N
N∑
n=1
1(−∞,a](Xn) = limN→∞
Xn ≤ a *'(5 1 ≤ n ≤ N *-
n3 ';1
N
6 2)12 6-+(', -23. 2) *--+11 5' *-(2-; 5'42 .-: 31 (+ 3'(85
FX(0),X(k)(a, b) = limN→∞
X(n+ k) ≤ b 7 (X(n) ≤ a *'(5
1 ≤ n ≤ N *-n
3 ';1N
6 ./, * +1 .--84 :(;5 . (::(5.3 (.1 -23.3 2) (2-;3 4 (/ ., +(122 .-: -+(', -23. '(5) ,-3 3:413-23.3 .,9 .1 (2 6* -2(+3 *-';13 4 (/1 .5 (: (9 3+5 ( 6-+ (', , (3
3.731 (+ 52 ) '
3.21 6X
2 3 (() +-1. -01. -',3 8(113 7) -+(', (:-, 195 -(2. (:-, X(n) = X, n = . . . ,−1, 0, 1, . . .
3.22
B(n).= B · (−1)n 6
g
k
( 3.22)
limN→∞
1
N
N∑
n=1
g(B(n), . . . , B(n+ k)) = limN→∞
1
N
N∑
n=1
g(B(n),−B(n), . . . , (−1)k−1B(n), (−1)kB(n)) 6
=1
2g(B(1),−B(1), . . . , (−1)k−1B(1), (−1)kB(1))
6 +
1
2g(B(2),−B(2), . . . , (−1)k−1B(2), (−1)kB(2))
6 = E [g(B(0), . . . , B(k))] .
6
A(n).= (B(2n), B(2n+ 1)).
6 A(n) = A(n+ 1)
6* (- .+(4:( 32/.3 .+(4 : * *2(, (', 19 4'; 2 .(. (, 2) 227 '+5 *3 3+:35 *-25(413 *-2+(136-; (T'(5 5('-45 (1 --4.-
( 3.26)(17 . (:(7.) -+7 4 -;1 (', ,(3 193 4';) *-/-:1 (:, 397 -23. '(56(7+ '57 :) - *, '51 . (;(. ) *-/-:1 * (:,
5 .5 3-3-) ) ('+:(Y (0)
(: -,'4, 3:.)1 '-+: 652 ) ' X(n), n = 0, 1, . . . -3 -3.23 - Y (n), n = 0, 1, . . . -23. .7 '-+: 6X(n), n = 0, 1, . . .
Y (n+ 1) =1
2Y (n) +X(n), n = 0, 1, . . .-+- 2
Z-,'4, 3:.)1 '-+: 6
Auto Regressive-23.
AR-23. ,'4: 397 -23.
Z =
∞∑
k=0
(
1
2
)k
X(k)
n'(5 *2(, 6(2-, *-1-,.1 32/.3 -,:.5 (. (, 2-/.: *, ,2, -':(-80 (:-, 227 '+5
Y (n)-23.3
(',3 .:(7. 2) 3,53 3'-3 ., *--41Y (n)
-23.3 (9 35-1 6Z
2) (2-;2 5('4Y (n)
2) (2-;3 2(+( 3.26)
.(-+lim
T→∞
1
T
T∑
n=0
g[Y (n)] = E[g(Z)]
6-+ (', ( -':(-80 5('-45 3-3- -23.3 4-;1 2(+N0
191 2/3 -23.5 :(5. : *, -+:3 5('-45
2) (2-;3 ., 5)/2 24 x3+(4:5 ,81: ' (7 -)3
t'5 *, 0';5 6-+12 0 (); 3:51
3.831 (+ '(7-) (2-32+ '(7-)3 '5 2 (21 39-, 3:)1 39 -, 39 5 () -/ '(82 6(:2 (+- +83 (2-;) * ()1 .,9
t + 1'5 (1 (416. (-5 (4'1 .,'4: (9 3:(7. 6x3+(4 :2
63+:35 ( 505 . (;(.2 *--01.1 *-2+(1 *. (-3 2)5 *--97'1 ( * -5 ()/ *-7-23. 2) 3/;)1 3 5(4'1 . (')')*-1 (/.2 31 (+ *3 3'(5 ( . (4)3 .(3:.3( 3 -/+2 * -1.-' (2, .('-) . (7'1 *-5)/1 .'()4. .(.)'*-/. (;1 * --01.1 *-27 ( 3-' (,. 5(4'1 . (')')2 '-) 2+(1 *. (-3 +521 6-97'1 -27 *3 *--5 (4'1 *-2+(1 *356) (1-)2 * -/ (:(
6++(5 .') * '(. , (3 * -5)/1 . (.)' 2) *-- -53 *-2+(13 +/,-',:-5
i.i.d.-23. 52 )' -23. -;2 *-)+/ *-)1.)1 *--1 +-/- ' (. 2, 6++(5 .') * '(.
4.1
*-)1.)13 27 ., .')1 +-/- .') 6λ3 (54 . ('5.35( -(2. -.25 ;(,5 (: )1.)1 -1 ' 275 '1 (27
*-)1.)1X(0)
(-30
'5) /-:: 6µ. ('5.35 -(2. -.25 ;(,5 (.+(5 ., *--1 *31 +/, 19 .+-/- 275 (6n';1 '5 .7'15 --+ '), *-)1.)13 ';1 .,
X(n)5 1 :(
+12) +8 (54 , (3 ,53 +83 (2-; 4-' (:-, '(.3 +( 27 '(7-) (2-32 +,1 5('4 ++(5 .') * '(.3*--.3( )1.)1 -1 ,2 *,
(−1)2) +8(
λ(1 − µ). ('5.35 '1 (27 .(') *--.1 ,2( )1.)1 -1 *,3'4 ,2) 225 *, -(:-) ,22 ',) : '(.3
(1− λ− µ) + 2 · λ · µ .('5.35 6µ(1 − λ) .('5.35 '1(27 .(')
-, 4-' '(.3 '),7 6λ · µ .('5.35 35-9 * ( 33 * (-3) * ()1 (,
(1 − λ)(1 − µ).('5.35 '5+
4-' '(.3 '),7) ,7 (:/:3 61−λ .('5.35
0( λ.('5.35
+1 3:() ,53 +83 (2-; 72( .(5-9 5 (17. ('12 6' (7 -) (2-3 (:-, 39 -23. 72 63,53 193 .+(4 :2 -.11 ,2, .('-) +--1 2542 2(7 - ,2 -1) /(42(:-,
(n + 1)'5 '(.3 2+( (2-;
n'5
X(n)2) '(. 2+( .:35 .(-5 (4'13 .:(7. 2 '1 () -23.3 .,93:() 4-' '(.3)7 5() -/3 -/7 (:3 5815 -(2. 5 () -/3 '(7-) (2-32 +(-:5 6'. (- 4 (/'3 '55 '(.3 '(,5 -(2.(-3 -(
Y (n)5n
'5 '(.3 '(, ., 1 : 3,53 3'(85 * '(.3 '(, ., 8--2 .-: 64 -' (:-, '),7 5() -/3131,.35 . ('-) * (- ( . (3 *-8--13 395 39 .5(
i.i.d.*--':-5 , .
A(n), D(n)
EA(n) = λ, ED(n) = µ . 6
3, (()13 ., *--41 '(.3 '(, -9,Y (n+ 1) = Y (n) +A(n) −D(n)I[Y (n) > 0] .
6 6,53 +85 (2-;3 . -54 '(82 35()/ 3:-, 3-' (033 -7 '('5 39 '(,.1
39 *.-'(2, '(5 * 63.13
31 (+5 ', (.13 -5-'(4'3 *.-'(2,3 , (3 . (-5 (4'1 .:(7. * '. (- -227 2+(1. ('5.33 -9, Y (n) = y
) (+- 2)12 *, 6Y (n+1)
2) (2-;3 ., 5)/2 ');, n
'5Y (n)
, (3 '3) .:35
227 3--(2. 3:-, (9 . ('5.3 -7 '('5 6y + g(y,X(n)) ≤ α) . ('5.33 ,-3 Y (n + 1) ≤ α '(,13 2)6,53 +83 (2-; 54: -/7 (:3 '3 (+-) '5 4(/'3 '552 +,1 24 -- '5+3 6* -+-+5 *-7' *-25413 *--5 (4'1 *-7-23.5 '1(27 .(-5 (4'1 . (')')5 4 (: 3 9 4';56. -:. (1 . ('5.3 2)
8.283'+35 '79: 63 :533
-23.3 6* -+-+5 * -7' 25413 +-+5 195 -,'4, -23. X(n), n = 1, 2, . . . -3 - 65 (4'1 .')')4.2
i, jn−k, . . . , jn( n > k272 *, 5(4'1 .')') ,'4-
P X(n+ 1) = i | X(n) = jn, X(n− 1) = jn−1, . . . , X(n− k) = jn−k
= P X(n+ 1) = i | X(n) = jn
3'+35 '), . (2.3 ' (/ '55 -(2. +-.5 (2-;3 -7 52 * -) : 6'55 -(2. (:-, +-.3 3 ((35 '3 .:35 '1 (2763 ((35 5813 .:35 4'( , ,-3-23.3 2) 5813 ,'4:
X(n)2+(3 6.')')3 2) '513 . ('5.3 ,'4:
P X(n+ 1) = i | X(n) = jn -(0 -53* 5(17 3;4. 3'+33 6S5 (. (, 1 :( 5813 5/'1 ,'4: .')')3 2) *--');,3 *-5813 (, 6
n'5-7' -7 227 '+5 /-:: * -:(1-5 . (-/ (: *)2 6* --5 (-/ ,4 (+ (,2( * -12) *-:1 92 (, *2) (:-,) +-+5 195 -23.26* -12) *3 -23.3
.-5(4'1 .')') '(5 * ) 52 * -)P X(n+ 1) = i | X(n) = jn, X(n+ 2) = l 6= P X(n+ 1) = i | X(n) = jn(4.3)
3.8
31 (+ '(7-) (2-3 Y (n) -3 - 6P Z(1) = 1 = p'),7 ±1
*-7' 25413 52 )' Z(n) -3 - 4.3 .(-(2. :-, -7 3,'3( '513 . (-('5.3 ., 5)/ 6* --5 (4'1 *3 *-7-23.3 -:) -7 3,'3 6Z(n) *-)';3 *63 97 (-(() * --41 (:-, '(7 -) (2-3 *2(, ( 4.3)
5 (-(() * --4.1 52 ) ' '(5 -7 3,'3 6 (+1 195-7 3,': (-2 (:2 ) -) ('/,3 +-15 4' -(2. -5(4'1 -23. 2) +-.5 (2-;3 3) 12
P X(n+ 1) = k | X(n) = i,X(0) = j = P X(n+ 1) = k | X(n) = i 6
7, ( 6+525 ('/,3 +-15 ,-3 . (2.3 '1 (27P X(n+ 1) = k | X(n) = i,X(0) = j =
P X(n+ 1) = k,X(n) = i,X(0) = jP X(n) = i,X(0) = j
6 P X(n+ 1) = k,X(n) = i,X(0) = j 6
=∑
in−1
· · ·∑
i1
PX(n+ 1) = k,X(n) = i,X(n− 1) = in−1, . . . , X(1) = i1, X(0) = j 6 =∑
in−1
· · ·∑
i1
P X(n+ 1) = k | X(n) = i, . . . , X(0) = jP X(n) = i, . . . , X(0) = j 6 =∑
in−1
· · ·∑
i1
P X(n+ 1) = k | X(n) = iP X(n) = i, . . . , X(0) = j 6 = P X(n+ 1) = k | X(n) = iP X(n) = i,X(0) = j 6
254:( 4.5)
5 5-8: 6. (-5 (4'13 .:(7. 225 .254.1 3:('/,3 -:;2 3'()3 '),7P X(n+ 1) = k | X(n) = i,X(0) = j = P X(n+ 1) = k | X(n) = i 6
* .(,'32 .-: '+ 3. (,5 4.4
6 P Y (n+m) = im, . . . , Y (n+ 1) = i1 | Y (n) = jn, . . . , Y (0) = j0
= P Y (n+m) = im, . . . , Y (n+ 1) = i1 | Y (n) = jn .
3:(7.3 7, (3((32( '52 /-5 .-'01- (9 3'+3 6 .5 *3 '53 ( +-.3 3((33 .:35 5 (4'1 .')')55(4'1 .')')5 :(5.: 4-(+1 5(1 (9 3+5(2 ..2 -+7 6193 '-8 ., *-7;(3 *, * .1--4.1 .-5(4'13.'+31 65 (4'1 .')') , (3 Y (n) * -7 /-7 (: 6Y (n) = X(−n)
3'+33 -+- 2 )+/ -23. '-+:( X(n).-:. (1 .2/(.
6 P Y (n+ 1) = i | Y (n) = jn, Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k
=P Y (n+ 1) = i, Y (n) = jn, Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k
P Y (n) = jn, Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k
=P Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k | Y (n+ 1) = i, Y (n) = jnP Y (n+ 1) = i, Y (n) = jn
P Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k | Y (n) = jnP Y (n) = jn
254:(( 4.12)
')45( X(n) 2) . (-5 (4'15 Y (n)
2) 3'+35 .7 )1.) : 6
P Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k | Y (n+ 1) = i, Y (n) = jn
= P X(−n+ 1) = jn−1, . . . , X(−n+ k) = jn−k | X(−n− 1) = i,X(−n) = jn
= P X(−n+ 1) = jn−1, . . . , X(−n+ k) = jn−k | X(−n) = jn
= P Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k | Y (n) = jn
254:(( 4.13)
5 .,9 5-8: 6X(n) 2) . (-5 (4'13 ., (:28-: 3:('/,3 3'()5 '),7 6
P Y (n+ 1) = i | Y (n) = jn, Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k
=P Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k | Y (n) = jnP Y (n+ 1) = i, Y (n) = jn
P Y (n− 1) = jn−1, . . . , Y (n− k) = jn−k | Y (n) = jnP Y (n) = jn
=P Y (n+ 1) = i, Y (n) = jn
P Y (n) = jn
= P Y (n+ 1) = i | Y (n) = jn
.,9 *7 : 6. -5 (4'1 .')') , (3 * Y (n) '1 (27 Y (n) .= X(−n) X(n)
4.5 Y (n)
P Y (n+ 1) = j | Y (n) = i = P X(−n) = i | X(−n− 1) = j P X(−n− 1) = jP X(−n) = i .
6
2) '513 . (-('5.3) 3'415 * -7 *-, (' (:, 6X
2) '513 . ('5.3 4 (-+5 , (3 2,1) +85 (),'3 -(0 -536195 . (-(2.Y
2) '513 . (-('5.3 195 )'(;15 . (-(2. :-,X6. -:. (1 . ('5.3 .'+31 .-+--1 5 (:
g =-3. 6
P Z(n) = −j = pj.('5.35 −1,−2, . . . ,−K *-7' 25413 52 ) ' Z(n) -3 - 4.6 -5-'(4'3 *.-'(2,3 -7 3,'3 61, 2, . . . ,K *-7' .25413 3-84 :(;
g(k, l)
Y (n+ 1) = Y (n) + g(Y (n), Z(n))6'513 . (-('5.3 ., 5)/ 6Y (n), n ≥ 0 -5 (4'1 -23. '8--1 3.13
31 (+ .(.)' 22(7 * -5' *-7-23. ( . (;(.2 * -2+(1 . (:52 .-: (.'95( -+12 -227 , (3 ('/,3 2-'.5 (:25-4) 3:5133,': ,2 -7 *, '5.1 6+( ( ' (8-- -7 -23. .(') . (7'1 3'(53 . (3:.3 '(5-+ . (. (, *-5)/1( .'()4.
, .-5(4'1 .')') -5 (4'1 -23. 2) 2+(12 2(4) 52 ) ' * -5-'(4' *.-'(2, 2) 2+(13) ,7 .,9-+- 2 ',.2 ');, -5 (4'1 -23. 27 ( -5 (4'1 -23. ',.1 *.-'(2, 27) 5 (15 *-12) ,4 (+ (,2 * -7' *652 , (3 ) '3 '),7 195 .)'(;1 . (2. 25 * -. -2 -5 -'(4' *.-'(2,((1 . (72 -+7 4 -;1 '-) 39 2+(1 (:-,') -;7 +/, +81 632 (;7 ,-3 5(4'1 . (')') 2) . (5-)/2 35-3*-5() -/ ');,2 -+7 4-;1 '+ (1 3:51 ) - 5 (4'1 .')')2 3,':) -;7 -:) +81 6. (7'1( * -7-23. 2) 5/'6.-'1 (: 3:-/51 . (/;2 -+12 30 (); 3'(85 *-)'(;1
3:('/,3 32532 6195 )'(;15 . (-(2. :-, '513 . (-('5.3 7 ( -;( *-5813 ';1 (5 3'415 3.1 97'.:* -1--41 7, *-:--:13 *-2+(13 5(' -:) +81( 35'35 * -0 (); * -) : * -5 () -/3 +--1 3,':) -;7 .(5- ';16-23.3 .:532 . (5' *'(. 39 (8-- 631',-+ (, .)' -+- 2 (97 . -5(4'1 .')') 8--2 .-: (:5 6(9 3:(7.6. (0 (); . (-'5 2, . (2(; -+- 2 852 ');, 39 3'415 *-(2-; 2) *-5 () -/ -7 3,'-:
n272 *, .-:(1 (3 .')') .,'4: 5 (4'1 .')')
4.7
P X(n+ 1) = j | X(n) = i = P X(n) = j | X(n− 1) = i
6195 . (-(2. :-, '513 . (-('5.3 *, '1 (27*-,53 *-25 (413 *-:(1-31 +/,5 )1.) : 39 3'415
P X(n+ 1) = j | X(n) = i = pij = p(j | i)
7 .'+ (1P'513 . (-('5.3 .8-'01 6
j2i1 '513 . ('5.3 . (, 3:7:(
P = piji,j=
p11 p12 p13 . . . p1N
p21 p22 p23 . . . p2N666 666 666 666 666pN1 pN2 pN3 . . . pNN
35 38-'01 -3 (9 6S = 1, 2, . . . , N '1 (27 61, 2, . . . , N *32 , ('42 (:'/5 (
N, (3 ,7 *-5813 ';1 '),7
“homogeneous .-:(1 (3 .')') /(:-13 6* -'5( (-2, 5813 ., 3+(13 ( -/7 (:3 5813 ., . 8--1 3'() 27
“stationary Markov .-':(-80 .')') (97 .')')2 * -,'(43 ) - 6-0'+:0 -'1 2 (:-,
Markov chain”)1.)32 +-;4 : (:, (9 35-1 -':(-80 -,'4, -23. /'735 3:-, .-:(1 (3 .')') 7) 301 39 / (:-1 6chain”6.-:(1 (3 .')') /(:-15
(, ( -.2/.33 (2-;3 -+- 2 -1)1 +/ ;(,5 54: .-5 (4'1 .')') 2) (2-;3 4 (/4.3
4';5 3,':) -;7.1',-+ '+ , (3 (: (8-- 6'513 .8-'01 ( -.2/.33 (2-;3 -+- 2 32(4) 3'(85 (, '513 . (-('5.36* -'513
'(.2 )1.)1 -1 *, 39 3'415 6. (1 (41 2)K
-; ( ';1 ) - '(.5) /-:: 4.1
31 (+2 )1354.8 3, (()13 .'95 8--2 .7 27 (: ' (.3 '(, ., 6'(.3 '(, 2 -;)1 (:-, 72( */: , (3 ,21 '(.3 '),7
Y (n+ 1) = Y (n) +A(n)I[Y (n) < K]−D(n)I[Y (n) > 0] . 6
3 '513 . (-('5.3 72pij = 0 |i− j| > 1
*,pi(i+1) =
λ i = 0*,
λ(1− µ) 0 < i < K*,
0 i = K*,
pii =
1− λ i = 0*,
1− µ i = K*,
(1− λ)(1 − µ) + λ · µ .'/,pi(i−1) =
0 i = 0*,
µ(1− λ) 0 < i < K*,
µ i = K*,
2) 3'(85 (2-, . (-('5.3 8--2 .-: -31 6. (5-9 ( . (3 -5) '+3 -5 2 3/:3 3:(10 (2-, . (-('5.35 6* -'51 .1',-+
0 1 K. . . .λ (1−µ) λ (1−µ) λ (1−µ)
µ (1−λ) µ (1−λ) µ (1−λ)
1−λ 1−µ1−λ−µ+2λµ
-; ( '(. 5 (4'1 .')') 6 '(-,'51 ',.1 &/ 27 65813 *) (7 (.5) 2(- -+- 2 8(-1 581 27 60, 1, . . . ,K *3 .')')3 -581 (9 31 (+56'513 . ('5.3 ., 8--13 ';1 (+-2 ( *-581 -5 -');,, (3 * -.183 ';1 (97 31',-+5 6. -:(1 (3 ( . -; ( 5 (4'1 .')') 272 *-'51 .1',-+ '--82 .-: 31 (+ 3'(854.8
31 (+1 .')')3 603:-, *. ('5.3) *-'513 ';17 , (3 . (.)43 ';1 6.')')3 2) *-5813 ';1763.-1 3+-2 -23. * .,'4:
*-,53 *-,:.3 ., . (1--41 .-:(1 (3 5 (4'1 .')') 2) '513 . (-('5.34.9
0 ≤ pij ≤ 1
∑
j
pij = 1 ∀i ∈ S .
61, (3 *-'513 .8-'015 3'() 27 -'5, * (7 '1 (27
(:18 ., ,81: +/, +8 '/,2 1.('5.35 -7 (0 -:)3 -,:.3 6. -:. (1 . ('5.3 .'+31 5 (: (),'3 -,:.3
7 -:)3 -,:.3 ., 5(.72 ');, -8-'01 (1 -5 6(3)27 5815P ·
116661
=
116661
61-18 ' *
P'513 . (-('5.3 .8-'01 2) -:1- -18 '(04 (( , (3
(1, 1, . . . , 1)T 3+(13 '(04 (( '1 (27
5811 '(5. .-5 (4'1 .')')) -(7 -3 (31 ( *-+8n'/,2
j5815 ,81. .-5(4'1 .')')) -(7 -3 (31. ('5.33 .--84 :(;2 (1 - '-+: .(-/ (:3 *)2 6. (245 5)/2 .-: .,9 -7 3,': 353 *-+8
n5
j5812
i6* -+8 ';15 '513 . ('5.32(6S = 1, 2, . . . ,K 5 .')')3 -581 ., 1 :( X(n), n = 0, 1, . . . .1--(1 5(4'1 .')') 54:
4.10 '(04 ( '-+: .7 6νk(n)
.= P X(n) = k '1 (27 νk(n)
5k5815
n195 .,81: .')')3) . ('5.33 ., 1 :3'()
ν(n) = (ν1(n), ν2(n), . . . , νK(n))
= (P X(n) = 1 ,P X(n) = 2 , . . . ,P X(n) = K)
5 *-+8n5j5812
i5811 '513 . ('5.3 ., 1 :
p(n)ij = P X(n) = j | X(0) = i
m > n 4.11
P X(m) = k | X(n) = i,X(0) = j = P X(m) = k | X(n) = i 6 ( 4.4)
n > nk > nk−1 > · · · > n1
4.11
4.12
P X(n) = j | X(nk) = ik, . . . , X(n1) = i1 = P X(n) = j | X(nk) = ik . 6
6. -;( ( . -:(1 (3 .')') 2) '513 . (-('5.3( (2-;3 5 () -/4.13
.-5-'(4'3 3, (()13 ., .1--41 *-+8n5 '512 . ('5.33 6
p(n)ij =
∑
k
p(n−1)ik pkj , p
(1)ij = pij
6* -'513 .8-'01 2)n3 349/3 '1 (27 p(n)
ij ij= Pn ,-3 *-+8
n5 '513 . (-('5.3 2) 38-'013 6
Chapman-Kolmogorov
5(' ((12(4 1; 8 ./ (: ., 72 (:25-4p
(n+m)ij =
∑
k
p(n)ik · p
(m)kj
P(n+m) = Pn ·Pm
-+- 2 5)/2 .-:n
'5 -23.3 2) . ('5.33 .--84 :(; ., 6νj(n) =
∑
k
P X(0) = k · P X(n) = j | X(0) = k
=∑
k
νk(0) · p(n)kj
ν(n) = ν(0) · Pn
6-' (04 (( (8--5 , (3 ('/,3 -(0 -53 '),7312)3 . ('5.33 ./ (:1 * ()': 63:03 ./7(3
P X(n) = j | X(0) = i =∑
k
P X(n) = j, X(n− 1) = k | X(0) = i
.-:. (1 . ('5.3 .'+31P X(n) = j, X(n− 1) = k | X(0) = i
= P X(n) = j | X(n− 1) = k, X(0) = i · P X(n− 1) = k | X(0) = i
. (-5 (4'13 225(P X(n) = j | X(n− 1) = k, X(0) = i = P X(n) = j | X(n− 1) = k
254: 72p
(n)ij = P X(n) = j | X(0) = i
=∑
k
P X(n) = j, X(n− 1) = k | X(0) = i
=∑
k
P X(n) = j | X(n− 1) = k, X(0) = i · P X(n− 1) = k | X(0) = i
=∑
k
P X(n) = j | X(n− 1) = k · P X(n− 1) = k | X(0) = i
=∑
k
p(n−1)ik pkj
3'+33 -;2) ((-7 6 3:0 ., /-7 (32 (:1 -- 75 (P = pijij
.,(()1 7) ((-7 6n272 3:(7: 3:03) 254: 3-84 (+:-,5(
P2 ,-3 * -+8 -:)5 '513 .8-'01 -7 -1 5(:. ('5.3 .'+35 )1.): - ., /-7 (32 -+7 6. (8-'01 2) 2;7 .(:(7.1 .-+--1 .5(: 5 ('( (12(4 1; 8* ()':( . -:. (1νj(n) = P X(n) = j
=∑
k
P X(n) = j | X(0) = k · P X(0) = k
=∑
k
νk(0) · p(n)kj
-'(04 (( (1 -5 (,ν(n) = ν(0) Pn
63/7 (33 ., (:1 -- 75 (. (-32 (:-2 -23.3 2) (2-;3 (31 .+2 -+7 6. (8-'01 2;75 *7.1 . ('5.33 .--84 :(; 2) 5 () -/3 7 *,4 (+5: .(8-'01 2;7 .(-'52, . (0 -)5 . () 2 .-: .,9 * *,3 6* --+1-1 5'3 *- (2-;3 27 ., 5)/2 * -2 (1
31 (+2P X(5) = i, X(100) = j
=∑
k
P X(100) = j, X(5) = i, X(0) = k
=∑
k
P X(100) = j | X(5) = i, X(0) = k · P X(5) = i, X(0) = k
=∑
k
P X(100) = j | X(5) = i · P X(5) = i | X(0) = k · P X(0) = k
= p(95)ij ·
∑
k
p(5)ki νk(0) =
(
P95)
ij
∑
k
(
P5)
kiνk(0) .
3,53 35()/3 3:413 .5 (: ,71 6-+11 5' (2-; 27 5)/2 .-: 31 (+ 3'(85195 (2-;3 -+- 2 -1)1 +/ ;(,5 54: X(n), n ≥ 0 .-:(1 (3 . -5(4'1 .')') 2) (2-;3 4 (/
4.14 6'513 . (-('5.3 .8-'01( 0
.()13 (2-;3 ., 5)/: 6*3)27 j1, j2, . . . , jk *-581 .'+ ( t1 < t2 < . . . < tk *-:19 .'+ '/5: 3/7 (3P X(tk) = jk, X(tk−1) = jk−1, . . . , X(t1) = j1
= P X(tk) = jk | X(tk−1) = jk−1, . . . , X(t1) = j1
× P X(tk−1) = jk−1, . . . , X(t1) = j1
= P X(tk) = jk | X(tk−1) = jk−1 · P X(tk−1) = jk−1, . . . , X(t1) = j1254:( 339 30 -)5 -)1: 6. (-5 (4'13 225= P X(tk) = jk | X(tk−1) = jk−1 · P X(tk−1) = jk−1 | X(tk−2) = jk−2
× · · · P X(t2) = j2 | X(t1) = j1 · P X(t1) = j1
=
(
k∏
l=2
P
X(tl) = jl | X(tl−1 = jl−1)
)
· P X(t1) = j1
=
(
k∏
l=2
p(tl−tl−1)jl−1 jl
)
· νj1(t1) =
(
k∏
l=2
Ptl−tl−1
)
jl−1 jl
· νj1(t1)
(2-;3 (.1 5)/2 .-:) (:-,' (. (, -+1 -1 +/3 (2-;3 ., *-+(- *, -+1-1 5'3 (2-;3 ., 5)/2 ');, '1 (276'513 . (-('5.3 ., ( '513 . (-('5.3 ( -.2/.33
ν = ν P 1 P
ν
4.15
3.17
0
ν(0) = ν
64.14
0;)1 ./7(31 . -+--1 72 .5 (: . -':(-80 6ν(n) = ν(n− 1)
-7 5 (: 5 (' ((12(4 1; 8 .,(()11 6-8-'013 (1 -5 , (3 -+-/-3 -) (43 6. (-; ( :-,) . (')') 2) 3'412 5-/'32 .-: (:-) ) *-5() -/3 ., 3'327 -9, 6. (8-'01 -5 *-2-'3 2;73 ( ' (5-/3 -227 ., . (1 --41 '), .(-; ( -, . (8-'01 2 5()/: *2(,6. (-;( ,2 . (')') '(5 * .(:(7 : . (:413 72( *-;4. (:-)) *-/ (. -;3
.(:(7. ( + ( - -) (1 317 2 +(12 ) - * -7 (', *-:1 95 5(4'1 . (')') 2) . (3:.33( 34 -1:-+3 ., -532 .:1 265 (4'1 . (')') 2) . (- -5';1 4' (5 '45: ,1) (, 5() ( 5 () 5812 '(9/: * -,3 (-'0 -'43 -;2 5(4'1 .')') 2) *-5813 ., ((2 .-:
*-1; 2) -; (6. -5 (-/ . ('5.3 .',.1 .)4 27 *2(, 6'513 . (-('5.3 . (1 ()' ,2 (:-:;2) 31 (+5
4.16
1 2 3
*-:) :( * -;2(/ * -581 6 '(-,(:'5) '1 6-.1 '('5 ,2 -7 *, /05(1 39 '51 72( 5812 '(52 -(7 - ) - +8 275 -9, 5815 2-/.: *,5() '(9/: ( *-5812 .,9 .1(2 6-,'4, -7 *, -; ( , (3 39 5815 3-3:) * -1;3 ';1 72 '(9/2 .-: ,265 () (24) 2 ' 275 *-'131 (:, ( *-24)
X(0)(:- -75) /-:: 6
Gambler’s ruin3 (, '131 2) (;(
4.17 -:;2 (:- -75) 73 * (7 X(n), n ≥ 0 -3 - 624) +-;: 339 . ('5.35( 24) /-((':1/2
.('5.35 6+/,04 2+53 * ,
4.131 (+ '(7-) (2-3 (, +-/- .') * '(.2 31(+ .-5 (4'1 .')') -3 (9 6
n';1 '(1-33
(-2, (:3 '),7 +/(-1 581 , (3“0”
581 '1 (27 6'132 +( 27 (: ,2( -(2-53 '1 - (:- -75) 73 '1 - '),76',) : *)';1 -7 '5+3 ) ('-; ,2 *, /8:2 )1- 4/)13) 7. - * -,3 6'('5 -'1 2 (:-, 5813 *-5813 ',)2 '),565 (17 -,'4, -;( , (3
0(:-,) 581 275 3-3:) * -1;3
(3)27 195i5812 ;, 195
j5811 '(52 . ('5.33 . (-32
ρji'-+:
ρji = P X(n) = i,(3)27
n > 0'(5 | X(0) = j
=
∞∑
n=1
P X(n) = i,X(n− 1) 6= i, . . . , X(2) 6= i,X(1) 6= i | X(0) = j
69
.('(,13) ((-7 * --4.1 (-(()3 6n
195 3'4i5815 (),'3 '(4 -53) ,(3 . ('5.3 *-5)/1 (:, (' (5 '(,136193 '01'; 2) *-:() * -7' '(5 *-'9 *3
(-2, 5() :) (:2 /05(1 i5815 32-/.1 .')')3 '),7 *,
Recurrent 3:) : 581 ,'4:
i581
4.18 6Transient
2(/ 581 ,'4: 3:) : (:-,) 581 6ρii = 1
*, '1 (27 +-.5 ./, *; . (/;26* -:) : * -5813 -:) 3'+33 -;2 6;-32( 5812 '(52 27 (: ,2 5815 2-/.: *,
4.19
1 2
*-:) : *-581 6 '(-,,2 39 3'415( (),' +85 39 581 5(9:) ;, 3:-,) . ('5.3 ) - 7) 2(/ 581 , (3
4.2'(-,5 5816*2(2 (-2, '(9/:
6* -;2(/ (-581 27 *, 2(/ -23. ,'4: -5 (4'1 -23.4.20
3,(()13 ., *--413 -23.X(n+ 1) = X(n) + 13;(.) '('5 6* -;2(/ * -5813 27 '1 (27 6*2(2 (-2, '(9/: ,2
X(0) = i(3)27 5815 2-/. : *, 7) 2(/ , (36(3)27 5815 ,81: -23.3 ' 275 7) -; ( ,(3 * -5813 ';1 *, 7.-. ,2 (97
* -'5+3 .(-5 (4'13 225 *2(, 6'/, 5815 (:2/.3 *, 2(/ 5815 *-'(4 -53 ';1 ., (:253 ,2 3'(,726* -' ()46Ni =
∑∞n=1 Ix(n) = i *--413 -,'4, 3:.)1 (39
i5815 -23.3 2) *-'(4 -53 ';1 .,
Ni5 1 :
j272 *, 4'( *, 2(/ , (3
i581
4.21
E [Ni | X(0) = j] <∞ 6 *-12) (-7') -5 (-/ 3:.)12 * --4.13
EN =∑∞
k=1 P N ≥ k ')45 )1.) : .2/(.3 5 () -/ '(82 72(
i5815 +/, '(4-5 . (/;2 (. (1)1
Ni ≥ 1.-),' 6
8.213:0 3,'
P Ni ≥ 1 | X(0) = j = ρji(4.21)
'1 (27i5 (: ' (4 -5 (-'/,(
m195 '1,:
i5 '(4-5 (1)1
Ni ≥ 2.7
P Ni ≥ 2 | X(0) = j
=
∞∑
m=1
∞∑
n=m+1
P X(n) = i,X(n− 1) 6= i, . . . , X(m+ 1) 6= i,X(m) = i,X(m− 1) 6= i, . . . , X(1) 6= i | X(0) = j
=
∞∑
m=1
∞∑
n=m+1
P X(n) = i,X(n− 1) 6= i, . . . , X(m+ 1) 6= i | X(m) = i,X(m− 1) 6= i, . . . , X(1) 6= i,X(0) = j
× P X(m) = i,X(m− 1) 6= i, . . . , X(1) 6= i | X(0) = j
=∞∑
m=1
∞∑
n=m+1
P X(n) = i,X(n− 1) 6= i, . . . , X(m+ 1) 6= i | X(m) = i
× P X(m) = i,X(m− 1) 6= i, . . . , X(1) 6= i | X(0) = j
=
∞∑
n=1
P X(n) = i,X(n− 1) 6= i, . . . , X(1) 6= i | X(0) = i
×∞∑
m=1
P X(m) = i,X(m− 1) 6= i, . . . , X(1) 6= i | X(0) = j
= ρiiρji. * 254: 3'(8 3. (,5 6ρ.'+35 ('/,5 ( .(-:(1 (35 (:)1.)3 ('/, -:;23 '515 '),7
P Ni ≥ k | X(0) = j = ρji · ρk−1ii
8.21
3:0 .'95 .2/ (.3 ., 5)/: 6ρii < 1
3'+33 -;2 2(/ 581 '(5 .7E[Ni | X(0) = j] =
∞∑
k=1
P Ni ≥ k | X(0) = j
=
∞∑
k=1
ρji · ρk−1ii
= ρji1
1− ρii<∞
(4.22)
9, (ρii = 1
3:) : 5813 *, .,9 .1 (2 6E[Ni | X(0) = j] < ∞ 2(/ 581 '(5 -7 (:/7 (3 6
ρii < 1'(1,7 7)63:03 ., (:/7 (3 75 6∞ .-
( 4.22)3,(()15 * (73
j = i'(5 .(/;2
X(k) 6= 3*,) 5 (: 5812 (, 5811 '(9/2 .-: ,2) ((-7 6
ρ33 < 1-7 3,': 2(/ , (3 581
4.1631 (+5-7 5 (:
ρ33.'+31 72 6
n > k272
X(n) 6= 3/'735 -9,
ρ33 = P X(1) = 3|X(0) = 3+ 0 < 1 6
62 (/ 581 (39 7, (
6j = i
'(5( 4.20)
3, (()15 -(0 -53 . (-; ( ., 4(+52 4 -;1 3'363,53 .-5-0 -, (0 :-,3 3,8(.3 ., . (,'32 -+7 (9 3,8(.5 )1.) : .7
6+/, 3:) : 581 . (/;2 32 ) - '1 (27 .;2(/ 3:-, *-581 2) -;( ';1 .25 .-5 (4'1 .')')4.22
19 + , (3)27 5815 * -'(4 -53 ';1 -7 52 * -) : 6* -:(1 -31 (. (, 0 -1) :( -.2/.3 581 (:54) /-:: 72 6, (3)27 5815 ,81: -23.3 ' 275 7)
n4 (-+5 , (3
019 22(7 ,2
n
∑
i
Ni =∞
-7 5(: .2/ (.3 . (:(7.1∞ = E
[
∑
i
Ni
]
=∑
i
E [Ni]
.-:19 (5 * --4.-) 7.- ,2 -; ( , (3 *-5813 ';1) ((-7E
[
∑
i
Ni
]
=∞, E [Ni] <∞ i272
62(/ (:-, +/, 581 . (/;2 72(6(5 *-' (4 -53 ';1 .2/ (. 5 () -/ -+- 2 2 (/ , (3 581 *, 4 (+52 .; (: 30 -) * .:. (: (9 3 :0
E [Ni | X(0) = i] =∑∞
n=1 p(n)ii 4.23
6∞ *-(() * --(0 -53 *, 2(/ , (3i581 *, 34-+52 .; (: 30 -) .:. (: 3:03
'79: 6E [Ni | X(0) = i] = ∞ *, 4'( *, 3:) -: , (3 581 .1+(43 3:03 -;2 -7 '79:(
i5815 2-/.: 5)/:( .:--81 3-84:(; 2) (1 -5
E [Ni | X(0) = i] = E
[
∞∑
n=1
1i[X(n)] | X(0) = i
]
=
∞∑
n=1
E [1i[X(n)] | X(0) = i]
=
∞∑
n=1
P X(n) = i | X(0) = i
=
∞∑
n=1
p(n)ii
6) ('+7
/-:: 6 5 (1 ./, 31-)1( , (1 ./, 31-)1 '. (-3 272 5)/12 -32 *-2(7 - ' 275) /-::4.24
(1 . (1-)12 '),7 ./, 31-)15 4 (-+5 2;01 5)/13 ' 275) /-:: 6. -0 -00 .(-(2. -.25 3 . (33)3'415 (2-,7 . (1-)1 .7'15 -, 7 *, ,2, .7'131 , (1 31-)1 ,8. ' 275 '1 (27 6. (;-+ ) - ,'(04 ((3 , (3 ,7 5813 6 5 (1 . (1 -)12 5)/13 3:;(- 39X =
( , (1 . (:-.11 . (1 -)1 ';15 (1 . (:-.11 . (1-)1 ';1).7'15 . (-32 27 (. +-.5 -9, .7'15 , (1 . (1-)1 ('. (: ,2 , (3)27 '5 *, -7 .7'13 '(,. -;2 '('54';5) 7.- 72 62(+ .(1-)1 ';1 * .7'13 .2(; ., (:2/.3) 7. - *2(, 63 9 (1 +/, 31-)1 '. (-3 2723'(831 581 27 *2(, , (1 . (1 -)1 ';1 .7'15 3:--3. -.2/.3 19
X =
(
km
)
62(/ 581 ,(3k > 1
'),7
, (3 * j581 -7 '('5 -9, 6*3)27
n,m'(5
p(m)ji = 1
7 (p
(n)ij = 1
-7 /-::( 3:) : 581 , (3i581) /-::. (:(7.2 *,.35 . (8(542 * -5813 ., ((2 .-: -7 3,': 39 4';5 6,) (:5 '. (- . (4 -(+1 . (,8(. ) - *2(, 63:) :6*32) . (:) -:3
j → i* (
i→ j*, 6
i→ j1 :(
j2 2-5 (1
i) '1,: -9,
i = j) (, , (3)27
n'(5
p(n)ij > 0
*,4.25 6
i↔j 1 :( communicating
*-') (41j(i) '1,:
60,-3 (182 58131 '513 . ('5.3 *, * (182 ') (41 581 27 3'+3 -;2 -7 52 * -) :
.1--4 *, 4'( *,i→ j
*-5813 .1',-+ .:-/51 6ρij > 0
*, 4'( *,p
(n)ij > 0
) 7n*--4 *+(43 (1 -5
(182i1 3'( 32-1 31',-+5 .1--4 *, 4'( *,
i↔j 71 3,8(.7 6j2i1 32-5 (13 31',-+5 32-16
j'+ .'5(3
1*-') (41
1, 5-7 . (,'2 -+--1 . (/; 6* -') (41
1, 4*-581 * ( *-') (41
1, 2*-581 (9 31 (+5
4.26
65812 *-2-5 (1
3, 7*-581(
35812 2-5 (1
2581 6*-') (41 *-; (: . ((9 (2-, 6 (7 (
1→ 4→ 5'+
52 2-5 (13,': )135 3:) :
1581 *,3 , 2(/ 581 , (3
7581 -7 '('5 6581 * ()2 *-') (41 *:-,
3, 6, 7*-581 *2(,67 2 0-2/32 +8-7
. ('-)43 /- . (:(7.4.27
reflexive relation.(-5-42;'
i↔i 6
1 2 3 6
74 5
*-'()4 *-581 6 '(-,symmetry
3-'01-j↔i *, 4'( *,
i↔j 6transitivity
.(-5-0 -9:'0i→ k
-9,j → k
* (i→ j
*, 66* -;2 (/ *3-:) (, *-:) : *3-:) -9,
i↔j *, 6m, k
*-1--4i↔j ) ((-7 6 ., /-7 (: 631 (+ 3,' .('+331 .-+--1 . (5(: . (:(),'3 . (:03 .) (2) 3/7 (3
) 7p
(m)ij > 0
* (p
(k)ji > 0
j'(5 (-'0 -'43 ., 4 (+5:( 3:) :
i) /-::
∞∑
n=1
p(n)jj ≥
∞∑
n=1
p(k+n+m)jj =
∑
n=1
∑
i1,i2
p(k)ji1
p(n)i1i2
p(m)i2j
≥∞∑
n=1
p(k)ji p
(n)ii p
(m)ij = p
(k)ji
(
∞∑
n=1
p(n)ii
)
p(m)ij
-7 (:-,' -9, 3:) :i*,
∞∑
n=1
p(n)ii =∞
* 72(∞∑
n=1
p(n)jj =∞
-9, 2(/j*, -:) +81 63 :) :
j-7 (:25-4 (
∞∑
n=1
p(n)jj <∞
* -9, (∞∑
n=1
p(n)ii <∞
6 3:(7. ., (:-,'3 75 62(/i* (
., 42/1 , (3 72 6. (2-4) /- , (3 72( -5-0 -9:'0 ( -'01- -5-42;' , (3 . ('-)43 /- .(:(7.1 3:416* -;2(/ *2(7) (, * -:) : *-5813 27 . (2-4) 38(54 275 3:(7. -;2 6.(2-4) . (8(54 2S5813 5/'1
4.28
n
p
(n)ij = 0
j i
*-5814.4
'(-, 2) 31 (+5 2)12 63'( 38(541 . (,8(-3 . (.)4 -, *-'513 .1',-+5 -7 5(: (9 3'+3163'( 38(54 5(17 , (3 *-5813 27 (, (:5 63' ( 38(54 *-((316, 7
27 .8(54 -3 (9 '1 (27 6. ('-)43 /-2 /-5 (2) . (2-4)3 .8(54 ,-3i581 2) . ('-)43 .8(54
4.29 6A
5 * -5812 4' 2-5 (1A
5 581 27 *, 3'( 38(54 .,'4:A
.('-)4 .8(54 6i2 *-') (413 *-5813
6* -:) -: 3-581 27 *, .-:) -: .,'4: .8(547581 2) . ('-)43 .8(54 6. ('-)4 .8(54 '-+32 27 (: '/5:) 581 272 6* -581
83,53 31 (+5
4.30 (:'. -, *(7-2 67581 ., * 32-71
6581 2) . ('-)43 .8(54 3-'01-3 225(
6581 ., * (+521 32-716(9 38(542 &(/1 581 ,2 * -2-5 (1 *:-, 6, 7 *-5813 7) 3'( 38(54 ,-3 (9 38(54 66, 7 .('-)4 .8(546'/, 581 ,2 ') (41 (:-, , (3) ((-7 .,9 +525 (. (, 32-713 3 .('-)43 .8(542 --)
3581 .,9 .1 (238(54 3::-, , .('-)4 .8(54 7, ,-3 1, 2, 4, 5 *-5813 .8(54 .,9 .1 (2 6
8581 '(5 * ;(. 39 /(. -:63 -2, --) (:-,)
35812 2-5 (1 38(542 --)3
2581 7) 3'(
1 2 3 6
74 5 8
*-581 ((- 6 '(-,) - -7 * -;2(/ 3-'5, 27 -9, 3'( 3:-, . ('-)4 .8(54 *, 6* -:) -: * -'5,3 27 3'( ( . -; ( .('-)4 .8(545638(/3 ( 3:11 -'0 - +/ '51
.; (: 38(54 ( . ('( . ('-)4 . (8(54 2) -;( ';12 4 ('-;2 .:.: . -; ( .')') 27 63/7 (3 ,22 4.31 63' ( 38(54 ,-3 ./, . ('-)4 .8(54 . (/;2 63'( 3:-,) 38(542 * -7--)3 * -;2(/ * -581 2) 34-' -2(, 6*-;2(/ 3'( 38(545 *:-,) *-5813 27 '(1,7 ( *-:) : 3'( .('-)4 .8(54 275 * -5813 27
. ('-)4 .8(54 , -,) 7. - -; ( /'735 ,2 . ('(3 . (8(543 ';1 *-581 2) 3-:1 5 ';1 * .')')563'( 38(54 ) - *, (2-;, 3:) -: 581 , -,) 7.-( 3'( 38(54 , -,) 7.- +/, 5811 '. (- .25,-3 .'/, 6./, 3'( 38(54 '),1 '. (- 32 ) - *, 34-'; .')') .,'4: .-5 (4'1 .')')
4.32 63'( 3182 0'; 38(54 .. 32 -, *,irreducible
.,'4: 38(54 6indecomposeable
34-'; ,2 .,'4:3,21 . (+-/, -, . (';5 634-'; ,2 ,-3 ,
irreducible3:-, +/, 2(/ 581 32 ) -) . -;( .')') -7 52 * -) :6
indecomposeable2 ,-3 3:((73 '),7
irreducible) (15 *-)1.)13 ) -( . ('+35
6* -') (41 *:-,) *-:) : * -581 -:) . (/;2 35 ) - *, 4'( *, 34-'; ,-3 .-5(4'1 .')') 3:41) - .-; ( 3'( 38(54 275 (+1 '-51 39 6(9 38(541 ,8: ,2 *2(2 3'( 38(542 --) -.2/.33 5813 *,3'( 38(54 .1--4) 7.- *2(, 6* -:) : 3 -581 27 -9, . ('-)4 .8(54 ,-3 38(543 *, ( .(/;2 +/, 3:) : 58138(542 &(/1 -.2/.3 5815 2-/.32 5(17 7.- 6397 , (3 -.2/.33 5813) ((-71 .,9 -: ,2 3-2, ( .; (:62(/ /'735 , (3 397 -.2/.3 581 6-,'4, *-+8 ';1 '/,2 3-2, -32( 3'(
62(/ 581 , (3 38(545 5812 2-5 (13 3'( 38(542 &(/1 581 27 63/7 (3 ,22 4.33
.(5-)/ 27 -, *2(, *-'513 .1',-+ ., /.:2 (:-2 *-:) -:( * -;2(/2 *-581 ((2 -+7) 7 *, *-, (' (:,+525 . (.)43 -+- 2 .(54: 3:) -: (, 2(/ 581 (2-, . (:(7. '513 . (-('5.3 2) *-4--(+13 * -7'2',) ( *-:) -: *36, 7
*-581 4'4.5
'(-85 2)12 7 60,-3 '513 . ('5.3 '),7 .)4 '--8: ,2 5 (17 '),7 6* -;2(/ *3 *-5813
+8-7 62 (+ 4-;1 *-+8 ';1 '/,2 58--.32 227 '+5 30 (: 5 (4'1 . (')') 2) (2-;3 -7 . ()/135 (:-,'(2-; * .-5(4'1 .')') *-2-/.1 *, -7 +--1 3,':
3.173'+3 .(-':(-80 2) ) (13 * (9 3+5 ( '5/26-':(-80 -23. 3-3. ,-3 -9, *-,.1
.-5(4'1 .')') '(5Stationary, invariant
-0 :, -' ((:-, (2-; (, -':(-80 (2-; ,'4 -:ν
(2-;4.34 ',)- -+1-1 +/3 (2-;3 39 (2-; * .')')3 ., 2-/.: *, '1 (27 6
n > 0272
ν(n) = ν''(
ν(0) = ν*,6 (54
-5 (4'13 3'415 *2(, 6-+1-1 +/3 (2-;2 4' (: , (3 25 (1 , (3 (:25-4) . (-':(-803 ( (),' 05154-;1 39 -:(1 (33
.-:(1 (3 5 (4'1 .')') '(54.35
.8-'01 2) -2,1) -18 '(04 (( , (3 '1 (27 ν · P = ν
*--41 , (3 *, 4'( *, -':(-80 (2-; , (3ν
(2-;3 661-18 ' * *-'513
.')')3 (:5 *, 6T
191 2/3 -':(-80 -23. ,-3 . -5(4'13 .')')3 -9, -':(-80 (2-; , (3ν(T )
*, 66-':(-80 -,'4, -23. ,-3 .')')3 -9, −∞ < n <∞ 19 272 .'+ (1-9,
ν · P = ν(ν(0) = ν
60 ('-; '.-5 3/7 (33 2 '(9/: 64.15
0;)1 3,' 3/7 (3ν(n) = ν(0) Pn
= (ν(0) P)P(n−1)
= ν(0) P(n−1)
= (ν(0) P)P(n−2)
= · · · = ν(0) P = ν(0)
3'+33 -;2 -9, -':(-80 , (3 (2-;3 *, -:) +81 6-':(-80 , (3 (2-;3 3'+33 -;2 72(ν P = ν(0) P
= ν(1) = ν
-74.14
0;)1 ./7(35 (:-,' 63'+33 -;2 4 (+5:(3.17
. (-':(-803 .'+35 '79: .7 6 3:0 ., (:/7 (3 75 (P X(nj) = ij , j = 1, . . . , k =
k∏
j=2
p(nj−nj−1)ij−1 ij
νi1(n1) . 6 2-2 (:,'3 (
(nj + τ) − (nj−1 + τ) = nj − nj−1) ((-7 .,9
τ5 *-:1 93 27 ., 9-9: *, 3:.)1 (:-, 39 -(0 -5'1 (27
n12) '5 -(2. (:-, 227
νi1(n1)-7
P X(t1 + τ) = j1 = νj1(t1 + τ) = νj1(t1) = P X(t1) = j1
./7(3 6T
191 2/3 -':(-80 , (3 -23.3 72( 6T
5(τ5 3(() 37' 0';5(
τ5 3-(2. 3:-, 2-2 . ('5.33 72('55 19 . (+(4: 2 .(27.3 -+- 2 -7 3+5(31 .5(: , -3 6,7 3. (, 2(27: ,2( '. (- .57'(1 3:('/,3 3:03'55 34 (/' 19 .+(4:1 -7 3+5(31( ;, ,-3 2(/ 5815 . (-32 . ('5.33 /'735 -7 . (,'2 ');, 4 (/'32-'.3 ., '(.; 3;(.3 .:532 6-':(-80 (2-;2 .-2,-8::(;4, . ('-315 :7.1 -+11 +/3 (2-;3 (2-; 271 (6,53
19 -:;2 * .(-':(-80 4-32 .-: ,2 -9,t1 < T
-; ( 191 2/3 .'+(1 .')')3 *, -227 ;(,5 -7 52 * -) :6. (,53 . (,1 (+3 . () -/11) -;7 T
−∞ < 0 4.36
n <∞
PX(−1) = 1 = 1
n = −1
1/2 1, 2
n < −1
PX(−1) = 1 = 1
n = −1
p12 = p22 = 1 1, 2
T > 0 n < −1
T
T
0 ≤ n <∞
−∞ < n < ∞
i
N Tr
P x(n) ∈ Tr | x(n−N) = i < 1. 6
n
P x(n) ∈ Tr = 0. 6
27 -'01- '(7-) (2-32 6-':(-80 (2-; -, .;2(/ .')')2 -7 '('5 +-/- , (3 -.1( -':(-80 (2-; *--4 -.13, (()13 2) (54 -+7 + +-/-3 -5 (-/3 ('.;3) ((-7 -':(-80 (2-; -, +/, 271/2
. ('5.35 ±1, (3 +8
.-;( -,3 ν · P = ν
6(2-; (:-, '),ν = (. . . , 1, 1, 1, . . . )
'(04 ((3 , (3(2-;3 *, 4'( *, 34-'; 3:-, .')')3 6-':(-80 (2-; ) - .-; ( 5(4'1 .')') 272 63/7 (3 ,22
4.37 6* -:) : * -581 2 4'01 *-:() * -7' 2541 -':(-803 (2-;3 6+-/- , (3
. (+-/- ' (/ 6+-/- -':(-80 (2-; 32 ) - --+irreducible
3:-, ,-3 *, * 34-'; ,2 .-; ( .')') -7 52 * -) :6'. (- (, .('( .(8(54 -.) * (-4 -+- 2 *':78
3,(()12 .-',:-2 3'52,5 . (,8(.1 5 (: -':(-80 (2-; * (-4 3'3Px = x
5(: 71 61-18 ' * -:1- -18 '(04 ( ) -
P38-'012 '1(27
x = (. . . , 1, 1, 1, . . . )T '(04 ((3 ('. -; ) -((-7 -7 /-051 . (8-'013 .'(.1 (-:5 ('; ('; 0;)1 6
1-18 ' * -2,1) -18 '(04 (( * ) - (9 38-'012)*-5-7'
P2) '. (-5 2(+3 -183 '2 --)3 -2,1)3 -183 '(04 ((2 -9, *--5 (-/ *3
P38-'013 -'5,)6-':(-80 (2-; (39
4.353:01( (2-; 3-3 -) 7 (. (, 21':2 .-: -; ( '(04 ((5 '5(+1) ((-7 6* --5 (-/
27) /-:: 6. (4 -'; -,3 .:(7. ., .1--41 3:-, 31 (+3 72( .('( . (8(54 -.) )- (:-:;2) 31 (+54.38 6
1/2.('5.35 *-'(4 *-'513
1 2 3 6 7
-':(-80 (2-; 6 '(-,(2-;3
0 ≤ α ≤ 1272 -7 4 (+52 24 -9,
ν = (α/2, α/2, 0, (1− α)/2, (1− α)/2) 6 ':(-803 . ('5.33 ., 42/2 .-: -5-0 -, (0 :-, ;(,5 6+-/- (:-, -':(-803 (2-;3 '1 (27 6-':(-80 (2-; , (3. (-('5.3 -+- 2 .54: 3'(3 38(543 (.5 . ('5.33 2) 34 (2/3 6(:-);: . ((,7 . (' (3 . (8(543 -5 . -638(543 (.5 '513*-58131 +/,2 '(5 : ,2 *2(2 9, (
25812 ,53 +85 '(5 :
1/2. ('5.35 -9,
35815 2-/.: *, 2-2 31 (+5, (3
22
11 '(52 -(7 -3 *2(, 6
1, 2, 6, 7*-58131 +/, 275 . (-32
1/439 3'415 , (3 -':(-803 (2-;3 6
6, 7*-5815 4' (,1, 2
*-5815 4' (, '5+ 2) (; (5 ,81: *--(1ω'(5 6
0, (3
62
11 '(52 -(7 -3) +(5
166, 7.(1)13 ., * ) -+32 5()/ .-5(4'1 .')')2 /-5 .(-':(-80 ) (15 ) (1 -)3 ., '-3532 5()/) *)763,8(.3 . 83 ( 3'+35 4;. : 72( '(43 '1 (/1 42/ (:-, ,) (:3 63 9 ')45 .(-+(', ) (13 2) .+/(-13
d i 4.39
n : p(n)ii > 0
6
d d 1
(indecomposable)
d
79
) - 3:(. : . ('-)4 .8(545 *-5813 272 '1 (27 .('-)43 .8(54 2) 3:(7. ,(3 '(9/13 '(, -7 . (,'32 .-:6*--'(9/1 *:-, *2(7) (, 339 '(9/1 1, 2, 3, 4, 5 4.40
pi(i+1) = 1, i = 1, 2, 3, 4, p51 = 1
d = 5
p12 = p23 = p34 = p41 = 1, p55 = 1
5 d = 4 1, 2, 3, 4
p12 = 1, p21 = 1/2, p23 = 1/2, p34 = p42 = p55 = 1
.-+(', .')') 2) (2-;3 *, -7 . (,'32 .-: *2(, 6-+(', -23. /'735 3:-, .-+(', .')') -7 52 *)'. (- 6-+(', -23. * ,2, -':(-80 -23. 4' ,2 ,-3 .')')3 -9, -':(-803 (2-;2 , (3)27 '5 3(()(2-;2
n → ∞ '),7 :7.1 X(n)
2) (2-;3 '1 (27 5813 (2-; -.2/.3 (2-; 272 .-+(', .')')5 71, (3 -0 :,-' ((:-,3 (2-;3 -52 (2-;3 -5 4/'13 *-+8t'/,2 '1 (27 -'01 (, , (3 . ( :7.33 584 6-0 :, -' ((:-,36* -'513 .8-'01 2) (2+5 -:)3 -183 '3 2) 02/ (13 '3 , (3
β( (54 , (3
C'),7
Cβt '. (-3 272
';1 , (3) 19 4+:-, * *--,'4, *-:.)1 .'+ '1 (27 +-+5 195 *--,'4, *-7-23.5 (:43
4';519 4+:-,5 4 ( : 39 4';5 6, -3)27 * -:19 .'+ ,(3) 19 4+:-,5 4 (2( (9 3'+3 5-/'32 ');, 6*2)-8/ *-';1 *3
T1, T2'1 (27 -;( . (-32 2 (7 - '),
, T1 ≤ t ≤ T219 2(('0 :-, '(5 '+(- -23.3 '1(27 6 -8'
-23. +-+5 195 (17 6T2 =∞ 7 (
T1 = −∞ -; ( :-, (, *3-:) ,2 ,T2 =∞ (,
T1 = −∞ '1 (27 -; ( :-,6* -:.)1 2) (, , (3 -,'4,a ≤ t ≤ b
X(t, ω)
[a, b] 5.1
., * ()':( 2913 '01'; ., 0-1) : *-.-2(X(t, ω), Xt(ω)
*-2(4)3 *-:(1-5 . (-/ (:3 -;2 )1.) : ,7 * 6X(t), Xt
7 -23.3X(t, ω), a ≤ t ≤ b 045 (54
ω'(5 6, 1 ,(3
X(2, ω),1 (+2 , 1 , (3
X(t, ω)2913 '01'; 2) 3-84:(;76 6 '(-8 3,' * +1 . -84 :(; .,'4:3
t2) 3-84 :(; , (3
ω2) *-:() * -7'5 * +1 . (-84:(; ) (2) 6 '(-,
-8' 195 *--,'4, *-7-23. 2) . (0 (); . (,1 (+ ';1 232* -(4 3 39 3'415 * +1 . (-84:(; .(- (;-0 2 .('(8 6, 1
A,B'),7
X(t, ω) = A(ω)t + B(ω) ,1 (+6* -') -
2)12 3-(2.3 .-,'4, 3+(0 -2;1,( '+)1 2) 32;33 195 2)12 3-(2.3 .-,'4, 39,; * ,) (: 2 ,1 (+23 . (' (8 6, 1A, φ
'),7X(t, ω) = A(ω) sin(2πft + φ(ω))
7 '(,.2 .-: *--24-9-; * -1'( 54 3./:356φ39,;(
A3+(0 -2;1, . (25 . (-:(1'3 . (+(:. 3 39 3'415 . (- (;03
3 39 3'415 . (- (;0 2 .('(8 6, 1 *3Xn
'),7X(t, ω) =
∑Nn=1Xn(ω) sinnt
-,'4, (, (54N
,1 (+6' (5 -+ . (,2 25(41 2+(1 (3 9 6. (-:(1'3 . (+(:. 2) 224 ()1 * (72) +525 -;( ';15 -(2. -,'4, -23. 27 ,2 0';5 63 97 0 (); '(,.2 .-: -,'4, -23. 27 ,2) 5 (17'1 (27 5('1 2
p(t)-3 - 63,53 3'(85 '+(1 -.'; .'()42 . (,2 25 (41 2+(1 31 (+2 6* --,'4, *-'01';
p(t).=
1 0 ≤ t < 1
0.'/, 6 6
2(;3 ., *-::;,13 ±1*-7' *-25413
An*--,'4, *-:.)1 *-0 -5 2) .-;( :-, 3'+- 2) '(+-) -9,-,'4,3 -23.3 ., *-:. (: 5('13
X(t) =∞∑
n=−∞
Anp(t− nT ) . 6
6T ≥ 1
'),7-,'4, -23. 2) . ('5.33 4 (/
27 (n27 '(5 142+7 . ('5.33 -4 (/ (,5 -- : 6-,'4, -23. Xt, T1 ≤ t ≤ T2 3-3 -
(T1 ≤ ti ≤ T2) t1, t2, . . . , tn
FXt1, . . . ,Xtn
(a1, . . . , an) = PXt1 ≤ a1, . . . , Xtn≤ an
0 - :(43 4 (/ ., *--42 5--/ , (3 (Xt
-,'4,3 -23.3 2) . ('5.33 4 (/ ,'4: (2-; . (-84 :(; 2) 39 (,.(-0 :
FXt1 ,... ,Xti−1,Xti+1
,... ,Xtn(a1, . . . , ai−1, ai+1, . . . , an) = FXt1 ,... ,Xtn
(a1, . . . , ai−1,∞, ai+1, . . . , an)
Poisson process(, (; -23.
.,-'4 (7 . (;(. 2) 5' ';12 2+(1 )1)1 39 -23. 6(, (; -23. ,'4:3 , . 2) 35()/ ,1 (+ 3. ,-5:*-5)/1 .'()4. .(7'15 +-1 '51 *-:(;20 .-97'12 . (/-) . -:7 ) -575 3+(4 :5 . (-:(71 '51 '-- 3:(1-,'4,5 )/'.13 (', (:182 ',.: (, (; -23. ., '-+32 .:1 2 6 (7 ( * -:('042, .0-2; *-'(. . (7'16
(0, t]193 4';5 * -)/'.13 . ('(,13 ';1 .,
Nt5 1 :( *; -+1
5.2
1
)1.) : 6+5250, 1, 2, . . .
*-12) *-7' 2542 2(7 -( +' (- ,2 -:(0 (:(1 , (3 (:--3+ 0 (); 3-:1 -23. , (3Nt
-23.3f(ε)/ε → 0
*,o(ε)
,(3f(ε)
-227 ;(,5 ∆t → 0
'),7∆t
-5 2 /-:9 2+( ,(3o(∆t)
,53 -01.13 (1 -5.(,53 . (/:33 ., /-:: -23.3 2) . ('5.33 4 (/ -5 2 6
ε ↓ 0'),7
5.3
(t, t+ ∆t] P(Nt+∆t−Nt) = 1 = λ ·∆t+ o(∆t)
λ ·∆t+ o(∆t)
lim∆t↓0
P Nt+∆t −Nt = 1∆t
= λ . 6
(t, t+ ∆t] P(Nt+∆t −Nt) = 0 = 1− λ ·∆t+ o(∆t)
1− λ ·∆t+ o(∆t)
3.8
N0 = 0
3,': )135 6-23.3 2) (2-;3 ., -1)1 +/ ;(,5 3'-+1 ,-3) '1(27 3'-5 3'+33) '('5 ,2 39 52)56.4;1 3'+33 7,)P(NT+∆t −Nt) ≥ 2 = P '. (- (, . ('(,1 -:) = o(∆t)
) 5(: 5 ( , 1*--( -:
n5 . (/283
k2 . ('5.33 -9,
(p+ q = 1) q(2) -7 -(7 - (
p3/283 -(7 - ) - ++(5 -( -:5 *, .'(79.-(0 -53 - 3:(. : * --(2. -.25
(
nk
)
pkqn−k =n!
(n− k)!k! pkqn−k .
*-042 193 '-8 ., 42/: '),7 2-2 -1 (:-53 . ('5.33 4 (/ 2) 2(55 254.- -23.3 6(, (; -23.2 '(9/:254:) *-'843 193 -04 '(5 619 -04
n = T/∆2 42/:
(0, T ]043 ., 63,53 3'(85
∆t'(,5 * -:04
NT = k4 (-+5 2542 .:1 2) 52 *-) : 6
p = λ∆5 ('-45 254: 04 04 275 72 6 , .(/:33 ., 2-;:
(', . (-32 -'8 19 -04k5 (2-, ( (2)7 (', ,22 . (-32 19 -04
n − k *-7-'8(0, T ]
045 *- (',72(
n→∞ 254:n = T/∆
) ((-7 ∆→ 0
04∆
'),7 (54k'(5 72 3/283
PNT = k ∼= n!
(n− k)!k! (λ∆)k(1− λ∆)n−k =n!
(n− k)!k!
(
λT
n
)k (
1− λTn
)n−k 6 =
n!
(n− k)!nk
(
1− λTn
)−k(λT )k
k!
(
1− λTn
)n 6 3:('/,3 3'()5 (),'3 -(0 -53 -7 .7 52 * -) :
n!
(n− k)!nk=n(n− 1) · · · (n− k + 1)(n− k)!
nn · · ·n(n− k)! =n(n− 1) · · · (n− k + 1)
nn · · ·n → 1 6
'1 (27 6n→∞ '),7
n!
(n− k)!k! → 1 .
(:5(
1− λTn
)−k
→ 1 ;
(
1− λTn
)n
→ e−λT
72(PNT = k =
(λT )k
k!e−λT .254: 3'(8 3. (,5 6(2-; (39 0';5 72(
λT'01'; * -:(, (; , 1 2) (2-; (3 9
PNT2 −NT1 = k =[λ(T2 − T1)]
k
k!e−λ(T2−T1)
-(0 -53 (PNt+∆ − Nt = 1 = λ∆e−λ∆ 254: 04
∆'),7 62-2 5 , .(/:3 . (1--4.1 7,) .(,'2 246'(2--0 5('-4
e−λ∆ ≈ 1− λ∆ -7 52 * -) : 5 3:0 '(82 612 , () -:)3
) '-) - (5)/ - /-7 (32 (, -:(, (; , 1 . (:(7.5 )1.)32 ');,E[NT ] = λT
E[N2T ] = (λT )2 + λT
Var(NT ) = (λT )2 + λT − (λT )2 = λT
) (1-) - .('-) - 32, . (,8(. /-7 (32 * (415 6-23.3 2) 5843 '01'; ,'4:λ
(+1 '-51 (),'3 (-(()3., 142+7 5)/: 63'+35 . ('-) - )1.)32 .-: +8-7 3,': *-:(, (; *--,'4, *-:.)1 '(5 .( (+- . (,8(.5(t0 = 0, tn = T ) t0, t1, . . . , tn
5 34 (2/3 . (+(4: ., 1 :( * -:04 *-042 42/:(0, T ]
043E[NT ] = E
[
n−1∑
i=1
(Nti+1 −Nti)
]
=
[
n−1∑
i=1
E(Nti+1 −Nti)
]
∼=n−1∑
i=1
λ(ti+1 − ti) = T · λ
E[N2T ] = E
(
n−1∑
i=1
(Nti+1 −Nti)
)2
= E
[
n−1∑
i=1
(Nti+1 −Nti)2]
+ E
∑∑
i6=j
(Nti+1 −Nti)(Ntj+1 −Ntj
)
= λT +∑∑
i6=j
(
λ(ti+1 − ti) · λ(tj+1 − tj))
= λT + λ2T 2
t2 > t1'(5 5,
E[Nt2Nt1 ] = E[N2t1 ] + E
[
Nt1(Nt2 −Nt1)]
= (λt1)2 + λt1 + λ2t1(t2 − t1) = λt1 + λ2t1t2
6 .5 *-)';3 -23. ,(3 * '(7-) (2-3 '(5 (:25-4)( 3.10)
3,8(.2 (9 3,8(. 3(()3
-9, t1 < t2 ≤ t3 < t4
*, (:--3+ .(-(2. -.25 . (; (. 2) -23. , (3 (,(; -23. 3/:3 225PNt2 −Nt1 = k1, Nt4 −Nt3 = k2 = PNt2 −Nt1 = k1PNt4 −Nt3 = k2254:
t1 < t2 < t3*-:1 9 4'1 '(5 72(
PNt1 = k1, Nt2 = k2, Nt3 = k3 = PNt1 = k1PNt2 −Nt1 = k2 − k1PNt3 −Nt2 = k3 − k2
254:(t0 = 0, k0 = 0
'-+:0 < t1 < · · · < tn
'(5 -227 ;(,5 (PNt1 = k1, . . . , Ntn
= kn = PNt1 = k1PNt2 −Nt1 = k2 − k1 · · ·PNtn−Ntn−1 = kn − kn−1
=n∏
i=1
e−λ(ti−ti−1) [λ(ti − ti−1)]ki−ki−1
(ki − ki−1)!
= e−λtn
n∏
i=1
[λ(ti − ti−1)]ki−ki−1
(ki − ki−1)!.
2) . ('5.33 4 (/ 72(t1, . . . , tn
272(n272
Nt1 , . . . , Ntn
2) . ('5.33 4 (/ ., 2542 *-2(7 - (:, (97 3'(85)135 67 '5+3 -, 227 '+5 60 (); , (3 . ('5.33 4 (/ *--(2. -.253 *-)';33 225 6 (+- (, (; -23.6- (, -23. -23.3 2) . ('5.33 4 (/ ., 0 (); ;(,5 ',.2 27 (: (5) '/, 3'41 3,':
Nt +Mt ν λ
Mt Nt
5.4
λ+ ν
X(n)
λ + ν Qt
5.5
Mt Nt
P X(1) = 0 = ν/(λ + ν) P X(1) = 1 = λ/(λ + ν) Qt
tn 1
N X(n) = 1
n n− 1 Q
tn N0 = M0 = 0
M
X(n) = 0
N,M
(, (; -23.2 * -' ()43 *-7-23..'9:2 5 ('4 -1 '-+:
Nt(,(; -23. '(5
Yε(t) ,Nt+ε −Nt
ε.(-84:(; 2) 3-8-9(;';( ,(3Y0(t)
) 254:ε → 0
'(5 E[Yε(t)] = λ(t+ε)−λt
ε = λYε(t) ≥ 0
) 52 *-)'-+:(ε > 0
2 '(9/: 64'-+Zε(t) , Yε(t)− E[Yε(t)] =
Nt+ε −Nt − ελε
-5 -/5:(Cov(Yε(t1), Yε(t2)) = E[Zε(t1)Zε(t2)]
5 -- : 6* --(2. -.25 * -)';3 2) *-7-23. *:-,Zε(t)
(Yε(t)
*-'41 -:)(5)/3 3/7 (3 ,22( 5 3'415 6
E[Zε(t1)Zε(t2)] = 0) '('5 , 3'415 |t2− t1| < ε
5 |t2− t1| ≥ ε ,
6 '(-85 3 -; (13 3,8(.3 .254.1 0 ();
6 '(-,t5 -(2.
(t > 0) PZε(t) ≤ α *,3 .(2,)t5 -(2.
(t > 0) PZε(t) ≤ α,Zε(t+ 5) ≤ β *,3
,5) ) ' ,-3 3,-8-3 27 ( 2:- -,) /-:: 6'513 .2;3 '1 2/3 *-'5 1 2) -; ( :-, (,1 3,-8-5 --:,53 * () -'3 ., 254: - (;-0 ;(,5 6(18 '5 131
6 '(-,6. -,'4, '51 .;(. 72 ',.1 ,(3 ( '513 . (11/.3 -23. ., ',.1
t = A193 .5-52 + (
t = 01 42/3602413 2) 3,-8-3 ) ' ., X(t, ω), t ≥ 05 ',.: 6+-1.1 581 7 '(,.2 .-:3 5812 '513 -1 71 '/,2
86
t < A'(5 (9 ,1 (+5 7, (
t5 -(2. 227 '+5
FXt(a)
.('5.33 4 (/ 6 (54a'(5
PX(t, ω) ≤ a 5 --:3-3-FXt
(a)) /-:32 '-5 +-1.1 581 2) 193 * (/.5 '1 (27
t A'(5 .,9 .1(2 6
t5 -(2. 4 (/3 3-3 -4 (/ 4' '(-83 3,'
t A'(5
t5 -(2. -.25
X(t, ω)) '5+3 ) ('; -, 52 * -) 6
t5 -(2. -.25 3) 12
) (13 ., '-+: +-1.13 5813 ., ',.13 193 * (/.5 . (-/ (:5 2;02 27 (:) . :1 2 6t5 -(2. (:-, . ('5.33
34'; +-+5 195 * -7-23. '(5 (:'+3) -;7 6 -':(-80 , . 2)
τ
t1, . . . , tn
n
X(t, ω),∞ < t <∞ 5.6
(
Xt1 , Xt2 , . . . , Xtn
)T
(
Xt1+τ , Xt2+τ , . . . , Xtn+τ
)T
FXt1 ,Xt2 ,...,Xtn(a1, a2, . . . , an) = FXt1+τ ,Xt2+τ ,...,Xtn+τ
(a1, a2, . . . , an)
.('5.33 4 (/ 193 '-8 . 9932 -0 :-' (:-, (2) . ('5.33 4 (/ *, -':(-80 , (3 , .) '1,: .'8(41 3'(85(::-, -24 - -; -23. 27) (02 ');, -':(-80 ,2 , .2 ,1 (+ , (3 (,(; -23. 6+525 * -:1 9 -)';35 -(2.63-89-2,-+-, ,(3 -':(-80 , . 2) ) (13 7, ( , (3)27 195 2-/.3 , (3) '/,1 -':(-80
n∆
X(n∆)
∆ > 0
X(t)
5.7
X ∆
.(,1 (+*-:((:1 , . *3) *-,53 *-7-23.3 -:)5 -- : 6
X(t, ω) = 5 , −∞ < t <∞
X(t, ω) = 5 sin 2πt , −∞ < t <∞
-':(-80 , (3X(t, ω) = A(ω)
-23.3 -9, ,(3)27 , 1A(ω)
*, (+1 6,2 -:)3 ( -':(-80 (),'3 -23.3 (+1
(+1 6-':(-80 (:-, (, (; -23. 6195 42/3 '),7 *-2-2) *-:195 ( -2-2) ' 25 (, (; -23. +( 4-5+: (,(; -23.2 3/7 (3 ,22 6(,
Y (t) , Nt+ε − Nt'-+: 6
Nt5 1 :
(−∞,∞)5 -23.3 ., 6* --(2. -.25 -2-2) 195 42/3 ( -5 (-/6* --':(-80 , . *3 (54
ε Yε(t)
(Y (t)
-9,Yε(t) , (Nt+ε −Nt)/ε
-,'4,3 -23.5 -- : 6X(t, ω) = A cos
(
2πf0t+ 2πφ(ω))
, −∞ < t <∞
[0, 1)*(/.5 +-/, ;(,5 2(;13 -,'4, , 1
φ(ω)( *--,'4, *:-, *- (54
f0, A'),7
6-':(-80 , (3X(t, ω)
-23.3 3:0634';5 .-1'(;-:(- 39,; * .-'(9/1 3-84:(; ' (5 .(-':(-803 .:02 3 (()3
52 * -) :(τ
54: 6α';13 2) *2)3 42/3 ., bαc5 1 : 6. -5 () -/ 3:-/51 30 (); 3/7 (3 .-),' 3/7 (3.-84:(; 2) . (-' (9/13 225 (:5 6
φ(17 2(;1 '1 (27
[0, 1]5 +-/, 2(;1
ψ.= f0τ + φ− bf0τ + φc , 13 -7
cos3
A cos(
2πf0(t+ τ) + 2πφ)
= A cos(
2π[f0t+ f0τ + φ− bf0τ + φc])
= A cos(
2πf0t+ 2πψ)
.
6-':(-80 -23.3 '1 (27 A cos
(
2πf0t+ 2πφ) (17 2(;1
A cos(
2πf0(t+ τ) + 2πφ) -7 (:25-4
142+7 '+(13B
'(,15 --:(a1, . . . , an, t1, . . . , tn, n
54: '. (- .0'(;1 .; (: 3/7 (3B =
ω : X(t1, ω) ≤ a1, . . . , X(tn, ω) ≤ an
142+7ΦB
., * '-+:ΦB =
θ : θ ∈ [0, 1)
⋂
A cos(2πf0t1 + 2πθ) ≤ a1, . . . , A cos(2πf0tn + 2πθ) ≤ an
', -9, Φ
2 --)3 397φ(ω)
(:2'3 *, '1 (27 φ(ω) ∈ ΦB
*,) 7θ*-';13 27 (, , (3
ΦB(:--3+
', , (3PB -9, [0, 1)
5 04 ,(3ΦB
*, +/(-15 (PB = Pφ ∈ ΦB -227 ;(,5 6
B(2 (:,'4) '(,13-9, * -;;(/ ,2 *-04 2) +(/, , (3
ΦB*,( 043
PB = ΦB., *-((313 *-043 -7', * (7
5.4
'(-, 3,' -τ. (175
Φ2) 39933 ., '-+:
[0, 1)5 ,-3)27 . (+(4: .8(54
Φ3--3.
TτΦ = φ : φ− τ ∈ Φ
* (/.5 +-/, 2(;1φ(ω)
( '/,1 .,9(Pφ(ω) ∈ Φ = Pφ(ω) ∈ TτΦ -9, [0, 1)
5TτΦ
* (Φ* *,) 52 * -)
X − [X ]1 :
X5 (
[X ] = 2, X = 2.3*, (7
X2) *2)3 42/3 .,
[X ]5 1 :
X-)11 ';1 '(5 6
[0, 1)
Φ. (+(4: .8(54 '(5 6
[0, 1)* (/.5 +-1.
X) 52 * -) 6
X = 4(
[X ] = −2-9,
X = −1.6*, 72 6,1 (+5 6
5.5
'(-, 3,' Φ = φ : φ ∈ Φ 38(543 .,
Φ5 1 :
(−∞,∞)5
-9,[0, 1)
5 . (+(4: .8(54Φ*, .7
P
φ(ω) ∈ Φ
= P
φ(ω) ∈ (Tτ Φ))
6 '(-,
6 '(-,2 (-)7 '(9/: 6+-/, 2(;1
φ-7 (7 : '5+3 5 () (
τ27 '(5
ΦB =
φ : A cos(2πf0ti + 2πφ) ≤ ai, i = 1, . . . , n
5 .7 -- :ΦC =
φ : A cos(2πf0(ti + τ) + 2πφ) ≤ ai, i = 1, . . . , n
72(ΦC = T−2πf0τΦB
-9,PXti+τ ≤ ai, i = 1, . . . , n = Pφ ∈ ΦC = Pφ ∈ T−2πf0τ ΦB
= Pφ ∈ ΦB = PXti≤ ai, i = 1, . . . , n
63/7 (33 ., (:12)3 .,95 (.-0 -:-1'0+ 3-84:(; 27 -7 . (,'1 3 72( (9 31 (+2 * --+(/--3 * -:(. :5 3)12 . ()1.)1 :-, . (/7 (33 -.)2)12 7 6*2) +/, '(9/1 2 +-/, ;(,5 . 2(;13 39,; 32 .-: *, -':(-80 , .2 (;3.
T'(9/1 * .-'(9/1, . ,(3
X(t+U)-9, 6
[0, 1]2 +-/, 2(;1 ( An5 .5 , 1
U3-3- 6
( 5.2)-+- 2 (. :3
X(t)-23.5 :(5.:6-':(-80
(Xt, Yt) Xt, Yt 5.8
τ
t1, s1, . . . , tn, sn
n
FXt1 ,Ys1 ,...,Xtn ,Ysn(a1, b1, . . . , an, bn) = FXt1+τ ,Ys1+τ ,...,Xtn+τ ,Ysn+τ
(a1, b1, . . . , an, bn)
N
6.()15 *-':(-80 /'735 *:-, * --':(-80 , . -:) -7 52 * -) :* -0 :1 (1
(+- *35 * -'413 6, .2 *-'()43 -:) '+1( (),' '+1 *-0 :1 (15 97'.:( . (-':(-803 ) (1 ., '2 5 (9:-:) ( (),' '+1 *-0 :1 (13 ., .+2 4-;1 . (-5 35'32 *2(, *-'-+: , . 2) . ('5.33 4 (/ . ()'(;14 (/ ., 3'-+1 -:) ( (),' '+1 *-0 :1 (13 . -+- (' (5) - (, -,'4, -23. 2) ) (13 *--4 3,':) -;7 7 ( 6.('5.33.('+3
.2/(.3 .-84:(;µX(t) , E[X(t)]
3-82'(4 (0 (,3 .-84 :(;RX(t1, t2) , E[Xt1Xt2 ]
:, -' ((43 .-84 :(;KX(t1, t2) , E
[
(Xt1 − µX(t1))(Xt2 − µX(t2))]
= R(t1, t2)− µX(t1)µX(t2)
.(0 (); . (:(7. ';1 *7 : 232 6* --,'4, *-7-23. -5 2 35' . (5-)/ )- 2 :3 . (-84 :(;2 )135 3,':) -;763 -82'(4 (0 (,3 . -84 :(; 2) . (5()/ (RX(t1, t2) = RX(t2, t1)
6RX(t1, t1) ≥ 0
638-'013
t1, . . . , tn27 (
n27 '(5 6
RX(t1, t1) RX(t1, t2) . . . RX(t1, tn)RX(t2, t1)666 666RX(tn, t1) . . . . . . RX(tn, tn)
6/7 (3 (+1 .-2-2) ,2 38-'01
6K
:,-' ((43 .8-'01 '(5 * -0'; 3'417 5 (17 . (:(7 : . (:(7.3 .) (2)5/'3 5(15 . (-':(-80
72t′(t272 ∀a FXt
(a) = FXt′(a)
*--4.1) '('5 -':(-80 , . '(5 .(-':(-802 '(9/:E[Xm
t ] = E[Xmt′ ]'1 (27 195 -(2. (:-,
µX(t)(),'3 0:1 (13 0';5 6195 * --(2. -.25 *:-3 , (3) '+ 271 * -0 :1 (13 72(
E[X(t)] = µX(t) = µX(0) ≡ µX*-0 :1 (13 7) -23.3 2) . (-':(-80 ''( '5+3 -, 195 *--(2. *:-, , (3) '+ 271 * -0 :1 (13 *, 52 * -)619 . (+ (4: ';15 .()13 (2-;3 2 +-1 * -4;1 *:-,*--4.32 5--/ -:)3 0:1 (13 -5 2 72(
FXt1 ,Xt2(a1, a2) = FXt1+τ,Xt2+τ
(a1, a2) (:5
E[Xt1Xt2 ] = E[Xt1+τXt2+τ ]
254:τ = −t1 '/5: *, (
RX(t1, t2) = RX(t1 + τ, t2 + τ)72
RX(t1, t2) = RX(0, t2 − t1) = RX(|t2 − t1|) (,RX(t1, t1 + τ) = RX(τ) = RX(−τ) = RX(|τ |)
* 72(KX(t1, t2) = KX(|t1 − t2|); KX(t1, t1 + τ) = KX(|τ |))';3 ,(3 ( +/, 3:.)1 2) . (-84 :(; 3 :, -' ((43 ( 3-82-' (4 (0 (,3 . (-84 :(; -':(-80 , . '(5 (:--3+6+525 *-:193
+525t1 − t25 -(2.
RX(t1, t2)(t5 -(2. -.25
µX(t)*, 5/'3 5(15 -':(-80 ,'4: -,'4, , . 3'+3
(:--3+µX(t) = µX(0) = µX
*--4.1t27 '(5 6,
RX(t1, t1 + τ) = RX(0, τ) = RX(τ)*--4.1
t2, t127 '(5 65
3-3 (2-,7 3,': , (3 -:) ( (),' '+1 *-0 :1 (13 .:-/51 *, 5/'3 5 (15 -':(-80 ,'4: , . .('/, *-2-15;-33 65/'3 5(15 -':(-80 * , (3 -9, *-1--4 * -:(),'3 *-0 :1 (13 -:) ( -':(-80 , . *,) '('5 6-':(-806* --':(-80 *:-,) 5/'3 5(15 *-':(-80 , . *:) -) . (,'2 ');, 227 '+5 (7 : (:-,, . ,(3
X(t + U)-7 (:-,' 6
T = 1'/5: . (0); *)2 6
( 5.2)(:;,3 -23.5 ( 5 ('1 2 2)
( 5.1)3'+35 '79:
Bn -3. 6-':(-80 (:-, '('-55 '), '/, . (,5 .7 :(5. : 6An5 .5([0, 1]
2 +-/, 2(;1U
*, -':(-80
'-+: 6U
5 .5(1
:,-' ((( ;, 8(11 * -21'(: (2-; -25 .5 , 1 .'+Y (t) =
−1∑
n=−∞
Anp(t− n− U) +
∞∑
n=0
Bnp(t− n− U) 6
.=
∞∑
n=−∞
Cnp(t− n− U) 6 (:-, . (,3 72( *--5 (-/ ( * --2-2) * -:19 ' (5 3:() 72 3-3- . (,3 2) (2-;3 6
Cn., '-+1 ('/,3 (-(()3 '),7
Cn) ((-7 6E Y (t) = 0
-7 . (,'2 24U
5 3--:.3 -+- 2 65/'3 5(15 -':(-80 ,(3 39 . (, -7 3,': 6-':(-80254:U
5 .5EY (t)Y (s) = E
∞∑
n=−∞
∞∑
m=−∞
CnCmp(t− n− U)p(s−m− U) 6
=
∞∑
n=−∞
∞∑
m=−∞
ECnCm
∫ 1
0
p(t− n− u)p(s−m− u) du 6 =
∞∑
n=−∞
∫ 1
0
p(t− n− u)p(s− n− u) du 6 -7 52 * -) : .7 6
1 :, -'(( ) -
Cn2) ((-7
p(t− n− u) = 1*, 4'( *,
t− 1 < n+ u < t 6
p(s− n− u) = 1*, 4'( *,
s− 1 < n+ u < s 6
*, 4'( *, ;,1 3:() 3--3. *.2;71 72t− 1 < n+ u < s , s− 1 < n+ u < t
6 24 . (-84:(;3 -.) 0 (0') -+- 2 6;,1 *-:()3 *-'5, -:) '. (-3 272 22(7 -; ( :-,3 * (73 6|t− s| < 1
'1 (276(t+1)−s = 1−|s− t| ,-3 * - (5-'3 -:) 2) 3;-;/3 -9,
t < s2)12 *, ( * --4.1 3;-;/3 -,:. *, -7 -7 . (,'23,8(.3 ., .-
t > s'(5 31 (+ 5() -/
E Y (t)Y (s) =
1− |t− s| |t− s| < 1
0 |t− s| ≥ 1
6 65/'3 5(15 -':(-80 -23.3 '1 (27
* --(2. -.25 , 1 '/5: .,9 * -+32 -+7 6. (-':(-80 '),1 5/'3 5(15 . (-':(-80 /-7 (32 '. (- 24 227 '+5-0 -:-1'0+A(ω)
'(5 6Xt = A(ω) cos(2πf0t+ φ(ω))
-23.5 :(5.: 6(0, 2π]
* (/.5 +-/, 2(;1φ) 7
A, φ5(15 -':(-80 -23.3) /-7 (: 6
φ5 -(2. -.25 -,'4,
A2 35/'32 .:.-: 3/7 (33 ( -':(-80 -23.3) (:/7 (3
(0, 2π]
* (/.5 +-/, 2(;1φ( . (-3 65/'3
E[Xt] = E
[
A(ω) cos(2πf0t+ φ(ω))]
= E
[
A(ω)]
E
[
cos(2πf0t+ φ(ω))]
= E[A(ω)] · 1
2π
∫ 2π
0
cos(2πf0t+ θ)dθ = 0
E[XtXt+τ ] = E
[
A2(ω) cos(2πf0(t+ τ) + φ) · cos(2πf0t+ φ)]
=1
2E
[
A2(ω) cos 2πf0τ]
+1
2E
[
A2 cos(2πf0(2t+ τ) + 2φ)]
=1
2E
[
A2(ω)]
· cos 2πf0τ + 0
39 0 (); -23. '(5) '(792 -,+7 65/'3 5(15 -':(-80 -23.3cosx = cos(−x)) ((-7 (
RX(τ) =E[A2]
2cos(2πf0τ)
6 .('5.35φ = 0, π/2, π, 3π/2
*;3 *2(, *-(2. -.25A
(φ*+(417 '),7 .,9 ,1 (+ 2 '(9/ 3'3
-':(-80 39 -23. *,3 5/'3 5 (15 -':(-80 39 -23. *,33-82'(4 (0 (,3 .-84:(; . (:(7.
6E[(Xt+τ −Xt)
2] = 2[RX(0)−RX(τ)]-9, 5/'3 5 (15 -':(-80
Xt*, 6
RX(0) ≥ 06
RX(τ) = RX(−τ) 66E[Xt+τ ±Xt)
2] ≥ 0.'95 /7 (3
RX(0) ≥ |RX(τ)| 63,':) -;7 ( -23.3 2) -:) '+1 *-0 :1 (13 ., .171 ,-3 3-82'(4 (0 (,3 . -84 :(;5 * -:-:(1 (:/:, (+1.-: 3-82'(4 (0 (,3 .-84 :(; . (1)1 2 +(,1 -42/ '53 6. (:-:1 . (2,) 2 .(:2 .');,1 ,-3 7 /,*--(2. -025 017
Xt+τ(Xt
+(,1 2(+τ'(5) /-::(
µX(t) = 0.(-':(-80 /-:: ,53 2(4-)31 . (,'239 3'415 6. -0 -00
RX(τ) −→|τ |→∞
0
27 (: 65.7
'(-,5 ', (.17 (,5.6
'(-,5 ', (.17 (3)1 3-3-RX(τ)
231 72(RX(0) ≥ |RX(τ)|) (:2 (+- 7 (17., 32, *-'415 ',.1
τ072(
τ > τ0'(5 .;,.1 017 3-82'(4 (0 (,3 .-84:(; 5('45) '1 (2 , (;-,
-503 ('793 2) '(,. -1 .:. (: 3-82'(4 (0 (,3 . -84 :(;) . (,'2 -+7 6-23.3 2) -5-04;,3 ('793
.(+(:. * .7 (+ 3-82'(4 (0 (, .-84 :(; 6 '(-,
. -:(0 (:(1 .7 (+ 3 -82'(4 (0 (, .-84:(; 6 '(-,. ('-313 *-1;
α,-3) . ('-315 302433 ., *-8-'1 (:, ( -0 :1 0' 2
X(t)., (:0243) /-:: -23.3 2)
254: . -' (413Y (t) = X(αt)
RY (τ) = RX(ατ)
|τ | → ∞ '),7 (+ (:-,RX(τ)
*'(5 *-7-23. * *:) -) '('5 6'84.1 ('793 ( &((7.1RY (τ)
-9, 2(+α*,(6. -,'4, 39,; * - (:-3 . (,3 2)12
.-84:(; 3. (, ., ) - * -7-23. -:)2 (5 581 7.- 6-23.3 2) +525 -42/ '(,. *-:. (: * -0 :1 (1) '(792 5()/5(15 -':(-80 -23. 7.-) (:-,' 0';5 6. -27.5 * -:() * +13 . (-84:(; 7 ( * -(2-;3 , 3-82'(4 (0 (, ( .2/(.6* +13 . (-84 :(; * 7 ( *-:() *3 *-:() * -:1 95 (2) *-(2-;3 '), 5/'3
.52081 3-82'(4
(9 ' (5 6*--,'4, *-7-23. 2) . ( (9 2) -:) '+1 *-0 :1 (15 * -' ()43 *-,53 *-) (12 44+9: )133 '(82142+7 3-82'(4 ('4 .520813 3-82'(43 ., '-+: Yt( Xt *--,'4, *-7-23.
RX,Y (t1, t2) , E[Xt1Yt2 ]
72(RX,Y (t1, t2) = RY,X(t2, t1)254.1
Zt = Xt + Yt'),7 6
t1, t2272
RX,Y (t1, t2) = 0*, 3-82'(4 -'/ *-,'4: * -7-23.3
RZ(t1, t2) = E
[
(Xt1 + Yt1)(Xt2 + Yt2)]
= RX(t1, t2) + RY (t1, t2) + RX,Y (t1, t2) + RY,X(t1, t2)
-9, 3 -82'(4 -'/ Yt( Xt *,) 5(17 6RZ
., 5)/2 * -8(' (:, '),7 -50 ;(,5 -; (1RX,Y
) 76RZ(t1, t2) = RX(t1, t2) + RY (t1, t2)-9, .()15 -':(-80 -':(-80 , (3 Xt, Yt,−∞ < t <∞ *--,'4, *-7-23.3 (9 *, 3'3
RX,Y (t1, t2) = RX,Y (t1 + τ, t2 + τ) = RX,Y (0, t2 − t1) 39 3'415 '-+:(RX,Y (t1, t2) = RX,Y (t2 − t1) (,
RX,Y (t1, t1 + τ) = RX,Y (τ) = RY,X(−τ)
-' (04 ((3 -23.3 (, .()15 *-':(-80 *-,'4: Xt, Yt,−∞ < t < ∞ *--,'4,3 *-7-23.3 (9 3'+3τ27 (
t27 '(5 *--4.1 *, 5/'3 5(15 -':(-80 ,'4:
E[Yt] = E[Y0]; E[Xt] = E[X0]6,
6 +525τ2) . (-84 :(; *3
E[Yt+τXt](
E[Yt+τYt],E[Xt+τXt]65
*3 , 5/'3 5(15 -':(-80 , (3 * -7-23. -:)1 +/, 27 (5 581 7. - '83 5(15 . (-':(-80 2) 3'415 (17, 1U
(T > 1
-3 - 65 ('1 2(; -+- 2 (:;,3 -23.5 '79: 31 (+2 65/'3 5(15 .()15 *--':(-80 *:-,-9, 6, 71/2
.('5.35 ±1*-7' *-25413 .5 *-:.)1 2) (, an, bn -3 - 6
[0, T ]2 +-/, 2(;13
*-7-23.31 +/, 27 -7 (:-,'X(t)
.=
∞∑
n=−∞
anp(t− nT − U) 6
Y (t).=
0∑
n=−∞
anp(t− nT − U) 6
+
∞∑
n=1
bnp(t− nT − U) 6
*2(, 6'83 5(15 -':(-80 , (3E[X1Y1] = E[a2
np(1− U)] = 1 6
E[X−1Y−1] = E[a−1b−1p(1− U)] = 0. 6 5(15 .()15 -':(-80 (:-, *-7-23.3 (9 . ('/, *-2-15 (, 5/'3 5(15 -':(-80 (:-, -' (04 ((3 -23.3 '1 (2765/'3
(Xt1 , . . . , Xtn)T -,'4,3 '(04 (3
t1, . . . , tn ∈ [a, b]27 (
n27 '(5 *, - (, ,'4: Xt, t ∈ [a, b] , . 3'+36- (, , ( , (3
3-82'(4 (0 (,3 ( .2/ (.3 . (-84:(; - .-1)1 +/ 54: (2) . ('5.33 4 (/ -9,[a, b]
5 - (, , .Xt
*, 3:0t, t1, t2 ∈ [a, b], RX(t1, t2), µX(t)
(2)-,'4,3 '(04 (3
ti ∈ [a, b]*-1--413
t1, . . . , tn27 (
n27 '(5 3/7 (3
Z =(
Xt1 , Xt2 , . . . , Xtn
)T
2) -:) ( (),' '+1 *-0 :1 (13 '(5 6- (, , . ,(3 Xt, t ∈ [a, b]) (:/:3 ( . (-3 - (, -,'4, '(04 ( , (3*--4.1
Z
E[Z] =(
µX(t1), µX(t2), . . . , µX(tn))T
(E[ZZT ] = E[Xti
Xtj] = RX(ti, tj)
- 3:(. :Z
'(04 (3 2) .-:-; (,3 3-84 :(;3 (ν1, . . . , νn)
27 '(5 72φZ(ν1, . . . , νn) = exp
i∑
νiµX(ti)−1
2
∑
i
∑
j
νiνj KX(ti, tj)
.-1)1 +/ 3'-+1 -,'4, '(04 ( 2) .-:-; (,3 3-84 :(;3 6KX(t1, t2) = RX(t1, t2)−µX(t1)µX(t2)
'(797 '),76-23.3 2) . ('5.33 4 (/ ., *-'-+1RX(t1, t2)
(µX(t)
72 -,'4,3 '(04 (3 2) . ('5.33 4 (/ .,6-':(-80 , (3 5/'3 5(15 -':(-80 - (, -,'4, -23. 3:41
*-7-23.3 * - (, , . Xt,−∞ < t <∞ *, 3:0a(t)Xt
a(t)Xt + b(t)Xt+c2(53 *,) * 3;8: 72 6/7 (3 (+1 *--0 -:-1'0+c, b(·), a(·) 27 '(5 *-- (, *-7-23.
dXt
dt= lim
ε→0
Xt+c −Xt
ε6- (, , . ,(3 * 2(5 3 .(1-,.1 . (/:35 -9, *--4-9, - (, , .
Xt3-3 -
∫ b
a
Xsds ≈∑
i
Xsi(si+1 − si)
*-7-23.3( - (, , 1 , (3∫ ∞
−∞
h(t− θ)X(θ)dθ,
∫ ∞
−∞
g(t, θ)X(θ)dθ
6* --0 -:-1'0+g(·, ·), h(·) '),7 *-- (, , . *3
6- (, -,'4, -23. .:. (: - (, -,'4, -23. 2 .-,'4, ,2 .-',:-2 32(; 3:41,53 2(4 -)31 2'0:, ( .2/ (. '+ -2/32 .-: * --:70 *-,:. ./. 3'3
E
∫ t
0
Xs ds ≈ E
∑
Xti(ti+1 − ti) =
∑
EXti(ti+1 − ti) ≈
∫ t
0
EXs ds
,(3 '+3 .;2/3 *)2 ) ('+3 -97'13 -,:.3E
∫ t
0
|Xs| ds <∞ 2(4)3 -,:.3 (, ∫ t
0
E |Xs| ds <∞ 6
6Y (t)
, (3 3,-8-3 -23. (X(t)
, (3 3 -:73 -23. 65.8
'(-,57 .7'1 3:(. :,2 (, .-',:-2 .-,'4, ,2 .7'1 3:(. : X(t),−∞ < t < ∞ -23.3 2) . ('5.33 4 (/ (+- 32,)3:-, 35().3 227 '+5 Y (t),−∞ < t <∞ 3,-8-3 -23. 2) . ('5.33 4 (/ (31 6 (+- 3:(-;,) .-',:-2
.-0 -:-1'0+ .7'15 . (, '51 6 '(-,2) . ('5.33 4 (/5 ,2 * -1; 35'3 --:.: -) 1 ;(,5 6195 3 (54 . -',:-2 .7'15 '5 (+1 *, (2-;, 3 (+-'5(+1 3-3 (2-,7 3,-8-3 4;3 39 2+(2 ,'4: '(8-43 12 6
E[Y 2(t)](,
E[(Y (t) − E[Y (t)])2]5 ,2,
Y (t).(, 02(43 0241 (. : *, 31 (+2 6+:2 '1:3 8(113 4;33 , (3E[Y 2(t)]
9, ( +/, *3(, 2) +: 2 /.15.;)3 ., /.:2 . (/;2 (, ,8(13 . (, (2-; ., /.:2 38': ) '3 .;)3 ., .-/;32 -+7 :7 (. , (3 ( ) ('5 () -/ 3:('.;5 4;. : 72( ('.; -, .-2273 3-52 6.7'13 ,8(15 ) '3 2) 4;33 5() -/ -+- 2 ) '36. -',:-2 .7'13 '),7(E[Y 2(t)]
.(,1 (+R-C
:1 ,
:1 6 '(-,*- (9 9 *2(5 5
*- (99 *2(5 6 '(-,2 .+2 -'8 31 (+- 32) (-;-,3) .-,'4, ,2 . -',:-2 .7'1 3:(. : 3,53 32,)3 ., / ::. -+-) '('5 6(9 32,)2 3,21 35(). 254: )135 6
E[Y 2(t)] = ., 5)/2 27 (:) .:1 2 X(t),−∞ < t <∞
142+7 .-' (-8 3'(85 ',.: 3-53 ., 6 (+1 34-;1 3:-,E[X2(t)]
6 '(-,- (. : 3.,-8-( .7'13 . -:7 -5 ')43
Y (t) =
∫ ∞
−∞
X(θ)g(t, θ)dθ ∼=∑
i
X(θi)g(t, θi)(θi+1 − θi) 6
*()':( -01.1 4 (-+ 2) . (-52 ,7 /-. : ,2E[Y (t)] ∼=
∑
i
E
[
X(θi)g(t, θi)(θi+1 − θi)]
→∫ ∞
−∞
µX(θ)g(t, θ)dθ
'+1 0:1 (13 ., (:254 7 6.2/ (.3 ( 3-8'0:-,3 '+ ., (:;2/3 ( 5.22
2 .2/(. (:85 .('/, *-2-15*-520813 *-0 :1 (15 .7 -- :
Y (t)2) (),'
RX,Y (t1, t2) = E
[
X(t1)
∫ ∞
−∞
X(θ)g(t2, θ)dθ
]
= E
[∫ ∞
−∞
X(t1)X(θ)g(t2, θ)dθ
]
254:( .2/ (.3 ( 3-8'0:-,3 '+ -2/: 5 () (RX,Y (t1, t2) =
∫ ∞
−∞
RX(t1, θ)g(t2, θ)dθ
8(113 3,-8-3 4;3 . -+- .22(7RY (t1, t2)
.-+-) 52 *-) 6RY (t1, t2)
5 --: (52E[Y 2(t)] = RY (t, t)
E
[
Y (t1)Y (t2)]
= E
[∫ ∞
−∞
X(θ)g(t1, θ)dθ
∫ ∞
−∞
X(η)g(t2, η)dη
]
= E
∞∫∫
−∞
X(θ)X(η)g(t1, θ)g(t2, η)nθdη
=
∞∫∫
−∞
RX(θ, η)g(t1, θ)g(t2, η)dθdη 6
.(:41. -+- .(-.5- ,2 (, . (-.5- 195 . (:.)1 (, 195 . ( (54 .(-,'4, ,2 . (-',:-2 . (7'1 '(5 , 6ta, tb
27 '(5RY (ta, tb)
.,(µY (t)
.-54 ., .');,1 −∞ < t, t1, t2 <∞ '(5RX(t1, t2), µX(t)4 (/ ., 3'-+1
Y (t)2) -:) ( (),' '+1 *-0 :1 (13 . -+- 72( - (, , .
Y (t)* - (, , .
X(t)'),7 5 63,-8-3 -23. 2) . ('5.33
6t1, t2
27 '(5RX(t1, t2)
., .+2 -'8RY (t, t)
., .+2 .:1 2) 52 * -)
,53 '(-85 *7 : 5.23
( 5.22
. (,8(.3 .,
.-',:-2 .7'15 '5 (3 -23. 2) *-0 :1 (1 6 '(-,
., 2542 72 27 (: 6(g(t, θ) = h(t − θ) '1 (27
h(t)*232 35(.3 - .:-; (,1 ,-3 195 3 (54 .7'13 '),7
-23. '(5 6) +/1 /.;: , 6. (1+(43 . (,8(.5h(t−θ) 5 g(t, θ) .;2/3 - 39 3'41 '(5 .()4 (513 . (,8(.3, (3 39 3'415 3,-8-( 3 -:7 -5 ')43) ((-7 6
g(t, θ) = h(t− θ)( µX(t) = µX-':(-80
Y (t) =
∫ ∞
−∞
X(t− θ)h(θ)dθ
3-8'0:-,3 ( .2/ (.3 '+ .;2/35 254:µY (t)
.= E[Y (t)] =
∫ ∞
−∞
E[X(t− θ)]h(θ)dθ = µX
∫ ∞
∞
h(θ)dθ
6195 -(2. (:-, , (3 * 35(.3 8(11) (:25-4 6. -',:23 .7'13 2)DC
'5 33 , (3 ('/,3 2'0:,3) 52 *-)3-82'(4 (0 (,2 '),5
RY (t, t+ τ) = E
[
Y (t)Y (t+ τ)]
= E
[∫ ∞
−∞
X(t− θ)h(θ) dθ∫ ∞
−∞
X(t+ τ − η)h(η) dη]
=
∞∫∫
−∞
E
[
X(t− θ)X(t+ τ − η)]
h(θ)h(η) dθ dη
=
∞∫∫
−∞
RX(τ − η + θ)h(θ)h(η) dθ dη
= [RX ∗h ∗ h](τ)
7 (17 6RX(τ) ∗ h(τ) ∗ h(−τ)2 '. (- (7 : (1 - ,-3 3:('/,3 3,(()13(
h(τ) = h(−τ) (:'+3 '),7RX,Y (t, t+ τ) = E
[
X(t)Y (t+ τ)] 6
= E
[∫ ∞
−∞
X(t)X(t+ τ − θ)h(θ) dθ]
6 =
∫ ∞
−∞
RX(τ − θ)h(θ) dθ 6 = [RX ∗ h](τ) 6 = RX,Y (τ) .
6 6( 5.21)
-,:. *--4.1 *, .'. (1 (9 3;2/3 (:'793) -;7 63-8'0:-, ( .2/ (. -5 '+ (:;2/3 (:-) ) *-/ (. -;56BIBO
.(5-8-2 2(4) 39 -, :. *2(,-9,
BIBO35-8-( 195 3 (54 . -',:-2 .7'13 5/'3 5(15 -':(-80 X(t),−∞ < t <∞ *, 3:41
( 5/'3 5(15 * --':(-80 X(t), Y (t),−∞ < t <∞
µY = µX
∫ ∞
−∞
h(θ)dθ 6
RY (τ) =
∞∫∫
−∞
RX(τ − η + θ)h(θ)h(η)dθdη 6 -':(-80 ( - (, Y (t) * -7 5(:( -':(-80 * , (3 -9, 5/'3 5(15 -':(-80 - (, , . X(t) *, 0';56.()15 *--':(-80 ( .()15 *-- (, X(t), Y (t) 7 (
9, (θ < 0
'(5h(θ) = 0
-9, .-.5- .7'13 *,RY (τ) =
∫ ∞
0
∫ ∞
0
RX(τ − η + θ)h(θ)h(η)dθdη 6
9, ( 3,-8-3 4;3 , (3RY (τ)
'(5 3,8(.3 2) -'4 -3 ) (1-)3RY (0) =
∫∫
RX(θ − η)h(θ)h(η)dθdη
6∞ + ( ;,1 (, ∞ + −∞1 *3 3-8'0:-,3 . (2(5 '),7 65/'3 5(15 * --':(-80 *3) 3 -:7 -7 -23.5( 195 . ( (54 . (7'15 4' 4 ( : 3,23 ( 3.1
3-'(; . ('1.3 195 . ( (54 . (-',:-2 . (7'1 (-;,2 . () - -.) ) -) (:+12 . (7'1( . (. (,5
X(t) =3-8-9(;';( -
X.-. ('-') 3 -:7 (8--(
h(t)*232 3.5(. - 193 * (/.5 .7'13 (-;, , 3)-6
Y (t) = X ∗ h(t) =∫∞
−∞X(τ)h(t− τ) dτ 35(.3 .254.1 6
X ∗ δ(t) =∫∞
−∞X(τ)δ(t− τ) dτ'+.3 , (3
2πf( '+.3 , (3
f,7
(ei2πft)-:(1'3 '('2
H(f)3.5(. - '+.3 * (/.5 .7'13 (-;, 5 3)-
X(t) =∫∞
−∞X(f)e2πft df37 (;3 3-'(; .'1.3 .(-:(1'3 . (+(:. * (7- -
X(t).-227 3 -:7 (8- 6-. -(93
, 3) - 3,-8-3 ( 3 -:73 -5 ')43 -0 -:-1'0+3 3'415 6Y (t) =
∫∞
−∞H(f)X(f)e2πft df3-8-9(;';( 5() (
, (3Y (t) =
∫ ∞
−∞
X(t− θ)h(θ)dθ
- (. : 3,-8-5 ( 3 -:75 3-82'(4 (0 (,3 . (-84 :(; -5 ')43 -,'4,3 3'415 (2-, (RY (τ) =
∞∫∫
−∞
RX(τ + θ − η)h(θ)h(η)dθdη
*)7( -');, 7, '5+3) 3,': )135 *--,'4, *-7-23. 2) 3'415 5 (1 ..2 ');, 5 3) -2 *,3 32,)6-,'4,3 3'415 * 7 '+.3 5/'15 39-2:,2 *-5' . (:('. - ) - -0 -:-1'0+3 3'415)63-' (; . ('1.3 3'9/
- .'+(1X(t)
2) 3-' (; .'1.3 6∫∞
−∞|X(t)|dt <∞) /-::( -,'4, ,2
X(t)5 2-/.:
X(f) = FX(t) =
∫ ∞
−∞
X(t)e−2πift dt
- 3:(. : 37 (;33 3-' (; .'1.3(X(t) = F−1X(f) =
∫ ∞
−∞
X(f)e2πift df
'1 (, 3-' (; . ('1.35 2(('; 0;)1 6(9 3 -51 *2.: *2(, ('/,3 2'0:-,3 * (-4 -52 3:-+ 3+(4 : .1--4 -9,i = 1, 2
'(5 ∫∞
−∞|Xi(t)|2 dt <∞ *,)
∫ ∞
−∞
X1(t)X∗2 (t) dt =
∫ ∞
−∞
X1(f)X∗2 (f) df
-':(-80 , . 2) - (;0 * + 2) 3-':,3 31 X(·) 3-84 :(;3 2) 3-':,3 ,'4: ∫∞
−∞X2(t) dt
2*--4 * -1-,.1 *-,:.5 -9,
FX(t) = X(f)*,
F
dX(t)
dt
= 2πifX(f)
-9, -)11X(·) *,
FX(−t) = X∗(f)
FX(t+ τ) = e2πifτ X(f)
FX ∗ h(t) = F
∫ ∞
−∞
X(t− θ)h(θ) dθ
= X(f) · h(f)
.-2'04; .(;-;8*, '('5 ,2 72( . -; ( :-, ,-3 .- (;-0 * +1 . -84 :(; 2) 3-':,3 -':(-80 , .
(X(t),−∞ < t <∞)'),7.-84:(; 2) 3-' (; .'1.35 --:.: (:, 6.,9 3) : ,2 (:, *:1, ( (97 3-84:(; 2 3-' (; .'1.3 852 ');, 2275
., Y (t), X(t) *--':(-80 , . '(5 '-+: 63 -82'(4 (0 (,3X(t),−∞ < t <∞ 2) SX(f)
.-2'04;3 . (;-;83 ,
SX(f) , FRX(τ)
.520813 .-2'04;3 . (;-;83 . -84 :(; 5 SX,Y (f) , FRX,Y (τ)63 -' (; .'1.3 ) -
RX,Y (·), RX(·) . (-84 :(;2) 3/:35 .,9(3,-8-5 . -2'04;3 . (;-;85 --:.: 6
E[X(t)] = 0/-::( 5/'3 5(15 -':(-80 X(t),−∞ < t <∞ (-)7 /-::
-/7 (:3 3'415 6. -)11h-7 /-:: (:, 6
5.13'(-,5 .',(.13 .-',:-2 .7'1 2)
RX,Y (τ) = E
[
X(t)Y (t+ τ)]
=
∫ ∞
−∞
RX(τ − θ)h(θ) dθ = RX ∗ h(τ)
35-8- 5 42 .7'15 ' 1 .(, '51 6 '(-, (+1 72(
SX,Y (f) = SX(f) ·H(f)
SY (f)
-5 2 632;712 37;3 3-8 (2 ((:(43RY (τ) =
∞∫∫
−∞
RX(τ − θ) + η)h(θ)h(η) dθ dη = RX(τ) ∗ h(τ) ∗ h(−τ)
3-8'0:-,3 '+ -(:-)5 72(SY (f) =
∫∞
−∞RY (τ)e−2πifτ dτ3'+33 -; 2
∫ ∞
η=−∞
∫ ∞
θ=−∞
∫ ∞
τ=−∞
e−2πifτRX(τ + θ − η)h(θ)h(η) dτ dθ dη =
∞∫∫
−∞
SX(f)e2πif(θ−η)h(θ)h(η) dθ dη
= SX(f)H(f) ·H∗(f)
= SX(f)|H(f)|2
'1 (27 6-)11h-7 3+5 (5 (:)1.)3 ('/, -:;23 (-(()5 '),7
SY (f) = SX(f)|H(f)|2
0';5 6'+.3 .5(. 2) 39,;5 ,2( 3+(0 -2;1,5 4' -(2. .-)11 .7'11 ,8(13 . (, * ('04;) (:254 (, (3 22(73 8(113 3,-8-3 4;3
RY (0) =
∫ ∞
−∞
SX(f)|H(f)|2 df
-4 (/) (:-,'(RX(τ)
2) 3-' (; .'1.3 ., -+12 .-. ('-') 3'(85 (:'+3 142+7 , (3 (-)72 (7 : 581363/ (: 3'(8 * -2541 195 3 (54 .-',:-2 .7'1 '+ '5133-. (:(7.2 52 * -) :
SX(f).-2'04; .(;-;8 ) (13 . (1)1 ., -532 -+7
.-'01- ( . -)11RX(τ)
) -:;1 .,9( .-)11SX(f)
7 (SX(−f) = SX(f)
,
6RX(0)
, (3 2,1)1 -(0 -53 -7 ∫∞
−∞ SX(f) df ≥ 05
6 '(-,6SX(f) ≥ 0
3:0
f0
-. ('-') '+. 5-5 -2,-+-, 0' '8 :1 . (-32H(f)
5 '/5: 3/7 (3'5 3-3- 39 3'415 3,-8-3 4;3
0 ≤ E[
Y 2(0)]
=
∫ ∞
−∞
SX(f)|H(f)|2 df ≈ 2SX(f0)∆f 6
6SX(f0) ≥ 0, f0
27 '(5 72(39 2+( 6f0
5-5 :13 5-.71) -;7 '8 *-'+. * (/.5X
-23.3 2) 4;33 ., 3) 12 ++(1E[
Y 2(t)]4;33 . (;-;8 4 (-+5 , (3 3-8'(; (';3 (54 ( 4-;1 '8 , (3 *-'+.3 * (/. '),7
∆f5 -',:-2 5 ('-45 , (3
(Amp)2/Hz(,
(Volt)2/Hz2) . (+-/-5 ++1:
SX(9 35-1 6. -2'04;3
'(797 6(0, 2π]
* (/.5 +-/, 2(;1φ( . 5
A, φ'),7
X(t) = A(ω) cos(2πf0t+ φ(ω))31 (+5 --:
RX(τ) =1
2E[A2] cos 2πf0τ
6 SX(f) =
1
4E[A2][δ(f − f0) + δ(f + f0)]
6 32,)5 4' ,2, 0'3 5/('5 -(2. (:-, 0' '8 :1 ,8(15 4;33( .227 (1 3-84:(; , -3 .(;-;8 3 (9 31 (+56,2 (, :13 -+- 2 */:
f0'+.3 *,
37-'8 3-84 :(;3 .,97 . (-32 -+7 3-82'(4 (0 (, . (-32 32(7 - . -(9 3 -84 :(; 27 ,2) 5(: (2, . (:(7.1 -7 52 * -) :6. -2-2) ,2 3-' (; .'1.3 .25 . (-32 2)12'5+2 .-: ,2 *, * (:--3+ *-:()3 * -'+.5 4;33 .2(7. 2) '(,. .(: . -2'04;3 . (;-;83 ) (1 *(7-22) .-2'04;3 . (;-;83 .-84:(; X(t),−∞ < t < ∞ -23.3 2) * +1 .-84:(; 2) 3-'(; .'1.3 2.-2'04;3 . (;-;83
SX(f)*, 7-;2 6'+.3 '-8 -;2 4;33 (2-; ., .:. (: 5/'3 5 (15 -':(-803 -23.3-9, -23.3 2)
2
∫ f0+∆
f0
SX(f) df
6(f0, f0 + ∆)
*(/.5∆
0' 5/' .25 . -2,+-,bandpass
.::11 3,-8-5 8(113 4;33 , (3* (/.5 3-':,3 .2(7. ., .+2 *-2(7 - (:, .-;( 3-':, .25 .-0 -:-1'0+ 3-84:(;
φ(t)'),7 52 * -)
4;33 .2(7.5 *-4 ( (:, ,7 6--:13 *-'+.3 * (/. 2 |Φ(f)|2 2) 2'0:-,5 (- - *-(1 *-'+.'-51 :1 '+
X-23.3 ., '-5: * +13 . (-84 :(; 2) 4;33 .2(7. ., ',.1 7,
SX(f)-7 /7 ((32 -+7-9,
ε(t) = X(t)−X ∗ h(t)7 3,-) -23. '-+: *, 60'2 &(/1SX(f) = 0
'),7 (−f0, f0) 0'
E ε2 =
∫
SX(f) |1−H(f)|2 df = 0 .
.('36FY (t) = 2πifFX(t) (' (5 *--4.1( * -3 (5 *-'+. 0 -251
Y (t) = dX(t)/dt'9( ,
,-3) *232 35 (.2 31-,.1Y (t) =
∫ t
t−∆X(θ) dθ
- (. :Y (t)
3,-8-3 (X(t)
3 -:73 -5 ')43) .7'1 5 (9 .7'1) +-1 5(: *232 35(.3 2) 3-'(; .'1.35 (- (.1 6* -'/, *-:1 95 ;, (∆
( ;, *-:1 93 -56*-3 (5 *-'+. .:'1 34-2/1.:1 2 -4;(,3 '-83 . (1)1 ., 4 (+52 ) - *35 *-2;01 (:,) (31 . (-55 4 (3 '(-8 2) 3'41 275 6'+. (, 19 ,051 , (3 *, ''52.-2'04; . (;-;8 /-2 ');, 6;, .2/ (. 25 -,'4, -23. '(5 .-2'04;3 . (;-;83 ., (:'+3 '(797 + .-2'04;3 . (;-;82 3-3. 39 3'415 6;,1 3:() .2/ (. -25 5/'3 5(15 * --':(-80 *-7-23.2 *
Y (t) = X(t) +C
.2542 X(t)
5 -(2. -.25C
, 1 - (:X(t)
, .2 *, (:--3+ 6;, '+.5 4'-+ . -84 :(;6SY (f) = SX(f) + E[C2]δ(f)
-9,
5.15'(-, 3,' .-2'04;3 . (;-;83
SX(f)3-3. 65/'3 5(15 -':(-80
X(t)-,'4, . (,5 4 (: 31 (+6α42/3 22(7 39 . (, 2)
) ' 22(73 . (, 2) .-2'04; . (;-;8 6 '(-,/-.1
α42/3 22(7
SX(f)) 3/:35 6(5 * -:-:(1 (::-,) ) '1 .5(:
α5 .:1 (13 .; (. 3) *-/-:1 (:,6
5.16'(-,5 ', (.17 3,'- - (;-0 * + - (, -,'4, -23.2
) ' 22(73 . (,3 2) * +1 . -84 :(; 6 '(-,5/'15 63'-9 852 * -)45.1 (:, 6
α5 '+.3 '() -15 1(13 42/31 (:--3 )'31 . (,5
X(t)2) .(+-' 3
3 '('5 39 .'7-: 3'(85 ) '3 04;, ., '-5 . 3'-93( |H(f)|2 = (2πf)25SX(f)
., 2-;732 (:-2 '+.3'+
X(t). (,3 .'53 - 342/3 852 8(1 ) '3 04;, 2 '5.32 -+75 6'+.3 5/'15 3( 193 5/'15', (.17 3,':
H(f) = Fh(t) 9, ( 5.17'(-,1
h(t)- .:.-: *232 3.5(.) 195 3 (54 .-',-:-2 .7'1
:13 2) *23 .5(. 6 '(-, (.1 6) '3 ., 4-2/: ,2 +(,1 04
T5 '/5: *, (18 2:-5 ;: +(,1 2(+
T5 '/5: *, 6
5.18'(-,5
:13 2) '+. .5(. 6 '(-,62-2 '(-85 -; (1
f1'),7
f1 = 1/πT) 7
T., '(/52 -,+7) 3,': *--:-: *-2(4 -)
;32 (, 3'-9 7 /, ( 342/3 852 *+(4 ) - (9 3-55 *,3 32,)6(18 .(,3 2) 3-' (; .'1.3 -52 3:-5 ')41 ( .-2'04; . (;-;8 2) ) (13 ., -532 (:2 '9( ,53 0;)13
1 : ∫∞
−∞|τR(τ)| dτ <∞ /-:: 0;)1
XT (t) =
X(t) if |t| < T
0 if |t| ≥ T
XT (f) = FXT (t) -9,E
[
1
2T|XT (f)|2
]
−→T→∞
SX(f)
193 2(('0 :-, '(,5 34 (2/3 6. -2'04;3 . (;-;82 (18 .(,3 2) 3-' (; .'1.3 -5 ')43 ., *-, (' ,764;32 3-':,1 (:. (, 3'-513/7 (3
E
[
1
2T|XT (f)|2
]
=1
2TE
T∫∫
−T
XθXηe−2πif(θ−η) dθ dη
=1
2T
T∫∫
−T
RX(θ − η)e−2πif(θ−η) dθ dη
72τ ≤ θ − η ≤ τ + dτ
(' (5) '83 ;3 2 3-8'0:-, *+(4 85: 5.19
'(-,5 --:(τ = θ − η 1 :
3-8'0:-, '+ -(:-) 6 '(-,
E
[
1
2T|XT (f)|2
]
=1
2T
∫ 2T
−2T
RX(τ)e−2πifτ (2T − |τ |) ·√
2dτ√
2
=
∫ 2T
−2T
RX(τ)e−2πifτ
(
1− |τ |2T
)
dτ −→T→∞
SX(f) = FRX(τ)(f) .
62 6) 61.520813 .-2'04;3 . (;-;83 ) (12 '(9/:
SX,Y (f) = FRX,Y (τ) = FE[X(t)Y (t+ τ)]6SWZ(f) ≡ 0
2(4) ;(,5 (, t1, t2
272 ;,2 3 (() '1 (27 EW (t1)Z(t2) ≡ 0
*, 3-82'(4 -'/W,Z
-7 '1,:65.20
'(-,5 ', (.13 3'415 +/(-15 --:
2-5415 . (7'1 -.)2 .;. ()1 3 -:7 6 '(-,
Yi(t) =
∫ ∞
−∞
hi(θ)X(t− θ) dθ
RY1,Y2(τ) = E
[
Y1(t)Y2(t+ τ)]
= E
∞∫∫
−∞
X(t− θ)X(t+ τ − η)h1(θ)h2(η) dθ dη
=
∞∫∫
−∞
RX(τ + θ − η)h1(θ)h2(η) dθ dη = [RX ∗ h1 ∗ h2](τ)
.-)11h'(5 72 6
h1(τ) = h1(−τ) (1 -5 (:)1.)3 '),7SY1,Y2(f) =
∫∫∫
RX(τ + θ − η)e−2πifτh1(θ)h2(η) dθ dη dτ
=
∫∫
SX(f)e2πif(θ−η)h1(θ)h2(η) dθ dη = SX(f)H∗1 (f)H2(f)
72( 3-82'(4 -'/Y2(t)
(Y1(t)
*--,'4,3 *-7-23.3 -9,5.21
'(-,5 (. :7 (7 H2(f)H1(f) ≡ 0
*, +/(-156*--(2. -.25 Y2(t),−∞ < t <∞( Y1(t),−∞ < t <∞ , .3 .()15 *-- (, *-7-23. 2) 3'415*3 *--,'4, *-7-23.7 *2(,
t(. (, '(5 3-82'(4 -'/
dX(t)/dt(X(t)
*--,'4,3 *-:.)13) /7(3 3'363-82'(4 -'/ ,2*-1-41 (2, *-::1 6
5.22'(-, -
H1(f), H2(f), . . .*-::1 2) (, '-+:
∀i 6= j, Hi(f) ·Hj(f) ≡ 0 ;∑
i
Hi(f) = 1
'+.3 .5(.5 3;-;/ ,22 * -::1 -:) 6 '(-,
'+.3 '() -15 3;-;/ ,22 * -::1 (, 6 '(-,:1 -+- 2 '5 (1 '+. 27 ( '/, '+. * (/. '-51 :1 275 7 '+.3 '() -1 ., *-42/1 *-::13 '1 (2731 (+31 6* -'+. -;2 . (,3 ., *-+-';1 * -::13 *-::13 272 2-5415 :7 :3
X(t).(, 2 .7 5 ()/: 6(3)27'(5
j:11 ,8(13 . (,2 /-5 3-82'(4 '/
i:11 ,8(13 . (, -7 4- : (-'/,) (-+3 (
5.21'(-, .1+(4363-82'(4 -, . (, 2) '+.3 -; -5 . ('/, *-2-15 6
u 6= j
.,-8-5 4;33 6* -::131 +/, 27 .,-8-5 4;33 ., 5)/2 *-2(7 - (:, .-2,'04;3 . (;-;83 ) (1 .'95(:,) ,-3 35()/3 3+(4:3 6:13 2) -97'13 '+.3 , (3f0
'),7 2 ·∆f · SX(f0)
'5 , (3∆f
5/ ('5 :12) 3,-8-3 -4;3 .+-+1 (.1 (+- ,2X(t)
-23. 2)SX(f)
., ')2( .,9 30 -)5 (;3 .722 * *-2(7 -'),7 +';: '+. * (/.5 +/, 27 0' -'8 . (. (, 2) (,2 . (,3 ., *-4';1 (:, (97 3'(85 62 :7 *-::1 '1. (-55 .-) (1) 30-) -3 (9 6+';:5 . (, 275 *-2;01( .5 - (, 3 3'415 72( 3-82'(4 -'/ *3 . (. (,363 -81'(;:-, . -/+5 2)12 . (5' . (-) 1
φ'),7
cos(2πf0t+ φ)5X(t)
5/'3 5 (15 -':(-80 , . *-2-;71)7 3'(4 31 3,53 3-55 4 (: 39 -5* -::;,1 ( '+)2 *-8(' (. (, - -53 . (,7X
2 5 ()/: 6[0, 2π]
*(/.5 +-/, 2(;1 (X(t)
-23.5 -(2. -.25 , 15 --: 6.' ()4. .(7'15 35' . (5-)/ (97 32(;2 6
cos3 .'95 (. (,
Y (t) = X(t) cos(2πf0t+ φ)
.(: 0 (); (5)/( 5/'3 5(15 -':(-80Y (t)
-9,RY (τ) =
1
2RX(τ) cos 2πf0τ
*--4.1g-3)27 .-0 -:-1'0+ 3-84 :(; ' (5 (+-7
F
g(t) cos 2πf0t
=1
2
[
G(f + f0) +G(f − f0)]
72(SY (f) =
1
4
[
SX(f + f0) + SX(f − f0)]
65.23
'(-,5 ', (.17 3(5 '+.5 . (, -+- 2 :; (,1 0' '8 . (, , 3'4165.24
'(-,5 ', (.17 3(5 '+.5 . (, -+- 2 :; (,1 *--(1 '+. 5-5 97 ('13 . (, 5 3'41
3+-+13 .7 '),7 -',:-23 (')3 .--5 2) . (. (,2 35/'3 -3 (9 6-',:-2 (:- 2) .-227 3-52 .7 '(5:.2/ (. *31 +/, 272 n(t),−∞ < t < ∞ ( X(t),−∞ < t < ∞ , . -:) *-:(. :) /-:: 6-23. ,-3t2
(t1
272E[X(t1)n(t2)] = 0)
3-82'(4 -'/ *-7-23.3) /-:: 7 65/'3 5 (15 -':(-80 *31 +/, 27 ( ;,'1 (27
n(t))'3 (
X(t)-(8'3 . (,3 * (7 ,(3)
Y (t). (,3 024: 65/'3 5 (15 .()15 *--':(-80 *3 72(.'53 -
X(t)-(8'3 . (,3 ., 2542( )'3 ., 422 *-8 ('
Y (t)024:3 . (,3 (.1 6
Y (t) = X(t) + n(t)(:- *-,'(4 (9 32 (;2 6
X(t)2 ');,3 277 3:1,: 3,-8- .254 (
h(t),-3 *232 (.5(.) -',:-2 :1 '+
Y (t)3( ) '31 3 .5 (:3 3,-) ) -( . (-3 -(8'3 . (,2 339 (:-, -(813 . (,3 6
5.26'(-,5 .', (.1 ,-3 ( ) '3 2) 225 6
5.27'(-,5 3-; (13 32(4)3 3'(85
5.26'(-, ., '--8: .,9 '-3532 .:1 2 6
h(t)- *':3 . (,3 . ((-1
:; (,13 . (,3 2) ( 0' '8 . (, 2) * ('04; 6 '(-,
:; (,13 . (,3 2) ( . (, 2) * ('04; 6 '(-,* ()'2 27 (:
n(·) -52X(·) -5 .-',:-23 . (2.3 -,
E
[
ε2(t)]
= E
[
z21(t)
]
+ E
[
z22(t)
]
5.27'(-,1 72(
E
[
ε2(t)]
=
∫ ∞
−∞
Sn(f)|H(f)|2 df +
∫ ∞
−∞
SX(f)|1−H(f)|2 df
) ' 2) ( . (, 2) .- (;-0 .-2'04; .(;-;8 6 '(-,
.-',:-2 .7'15 *-'5( ) ' .; (.5 . (, 6 '(-,
.-',:-2 .7'15 *-'5(3 ) ' .; (.5 . (, 2) 32(4) 3 83 6 '(-, 62)12 6
H(f)2 . (:() . (83 . (()32 27 (: (-)7 6. 22(73 .8(113 .- (5 -'3 3,-)3 '(5 )'(;1 -(0 -5 (:254
852 f0
'01';3 2) 3-84:(;7 3,-)3 ., 5)/2 27 (: R-C
.::1 (:--3+ H(f) = (1 + if/f0)
−1 '(563 97 :1 '(5 .-21-:-1 .8(11 .- (5 -' 3,-)2 * ('-) '. (-5 5 (03f0
'01';3 ., ,(812( 3-89-1-0;(,3'415 6* (1 -:-12 .8(113 . - (5-'3 3,-)3 ., ,-5-) -21 -0;(,3
h(t), (3 31 2(,)2 ,-3 '. (- . 9 (: 3) -
-21 -0;(,3h(t)
.,-81 2) *-'41 -:) -5 -/532 ) - 396-.5- 3-3-
h(t)) 3)-'+ -, (:--3+
h(t)2 . (253 -, 6,
6-.5- 3-3 -h(t)
) 3251 .1--4 65(:-3 .-5 6- (2:, ;(,5 *, ( -20 --+ ;(,5 *, (2) 5 ('4 (, (. (, )112 .-: -9, -.5- , (3 ('.;3 *,(. (, ,-5: ,2( 3)4 ,(3 ('.;3 *2(, ':-( . ::17 3 (+-(
Wiener- 3'.;: . (-.5-3 .25 1 * -21 -0;(,3*,3 ,-3 +-1 .2,) :3 32,)3 6)135 (5 97'.:( . -/- 24 ,(3 . (-.5-3 .25 1 ,22 ('.;3 639 '(45(, -(2:, ;(,5 (2) 5 (0 5('4 (, (. (, )112 .-: *,3 (:--3+ -24 -9-; 5(1 ) - . (-.5-3 .251 ,22 ('.;23:-,
h(t)-9, /-:9 (,
h(t) = 0, t < −T0'(5 '),7
5.28'(-,5 3 -; (13 3'(831 ('.; (:254 *, -20 --+
-; ( -2-2) 42/ * .-.5 ,2 *23 .5(. 6 '(-,(7 3--3)3 2(52 ');, 35 . (-55 72 6)112 27 (:
5.29.7'13 ., '1(27 6. -.5-
h(t−T0)*2(, . -.5-
3-3)3 -+- 2 .-.5 ,2 .7'12 -) 1 5('4 6 '(-,':-( . ::1 *-,'(4 39 ('.;2 6. -24 -9-; . (1)1 . (-.5- ,22 ('.;2 ) - 3'45 . (-52 +(-:5 .'()4. . (-56.-;( :-, 3--3)3 *
.- (5 -'3 3,-)3 6* (1-:-12 .8(113 . - (5-'3 3,-)3 ., 3,-513 .-21-0;(,3 .::13 .,-81 . -52 '(9/:- '(1,7 .:.-: .8(113
E
[
ε2(t)]
=
∫ ∞
−∞
Sn(f)|H(f)|2 df +
∫ ∞
−∞
Sx(f)|1−H(f)|2 df
6H(f)
.(::13 27 -:; 2 * (1-:-1 3-3-E[ε2(t)]
) 7H(f)
.::1 ,81 3-53(,'4:3 * (/.5 -(81 -227 ('.;( .57'(1 ,-3 * (1-:-12 2'0:-, 3,-513 3-84 :(; .,-81 2) .-2273 3-536* -52) 3) (2)5 3. (, '(.;:( .-/- 30 (); (:-:;2) 3-53 (:2912 6. (-8, -' (( (5)/
+/, 2'0:-,7 3,-)2 -(0 -53 ., * ()': .-),'E
[
ε2(t)]
=
∫ ∞
−∞
[
Sn(f)|H(f)|2 + Sx(f)|1−H(f)|2]
df 6
3-84 :(;2 ./.1 /0)3) 7H(f)
*-);/1 (:, 7 *, 6* --5 (-/ 2'0:-,3 ./. *--(0 -53 27 -7 52 * -) :(Sn(f)|H(f)|2 + Sx(f)|1−H(f)|2(:-:;2) 3-55 -, +';:5
f2) ' 275 3-84 :(;3 ., -0432 (:-2 39 /0) -0432 -+7 *2(, 6-21 -:-1 3-3-.,-812 2'0:-, 2) * (1 -:-1 .,-811 3-53 ., (:0) -; '1 (27 6* -:() * -'+.5
H(f)-7' ., *-') (43 *-8(2-,63-84:(; 2) * (1-:-1
5-5 213 2 .(+(4 :3 27 -5) 52 * -) 65.30
'(-,5 .,9 00') : 6|1 −H(f)|5 -- : * -(1f'(5 -:) 52)
H2) 39,;3 .'-/5 6 '(-,
-2/:f(. (, '(5 *, 72 6|H(f)| .-)113 3+(4 :3 ,-3
13+(4 :2 '. (-5 35('43 |H(f)| = const
.-),'3-'52, ;(,5 6|1 −H(f)| .:0432 * (': |H(f)|5 H(f)
.,|1−H(f)| ≥ 1− |H(f)|
|1−H(f)| ≥ |H(f)| − 1
72(|1−H(f)| ≥ |1− |H(f)|| .2-532 '. (1 72(
SX(f)|1−H(f)|2 -:)3 '5-,3 ., -04 : 25,Sn(f)|H(f)|2 -(0 -53 ., 3:) : ,2 (9 3;2/3 -6
f27 '(5 .(-2-2) ,2( . (-)11 3)
H(f). (-84 :(;2 ) (;-/3 .,
-(0 -53 3'(5) .-2-2) ,2( . -)11H(f)
.,-812 72 .181081 3-53 -) -2) 52)Sn(f)
[
H(f)]2
+ SX(f)[
1−H(f)]2
-+- 2 * (1-:-13 ., ,81: 6-21 -:-1 +:'0:-,3 (' (5H(f)
(. (, .,a = H(f)
5 (. :f'(5 1 : 6-21 -:-1 , (3
∂
∂a
Sn(f)a2 + SX(f)(1 − a)2
= 0
254:(a =
SX(f)
SX(f) + Sn(f)
72(Hopt(f) =
SX(f)
SX(f) + Sn(f)
6 .8(113 .- (5'3 3,-)3 ./ (: (.2
Hopt(f). 83 - .254.1 .-21-:-13 3,-)3(
E
[
ε2min(t)]
=
∫ ∞
−∞
(
Sn(f)S2X(f)
(SX(f) + Sn(f))2+
SX(f)S2n(f)
(SX(f) + Sn(f))2
)
df
=
∫ ∞
−∞
SX(f)Sn(f)
SX(f) + Sn(f)df
2) 02/ (1 . (/-:2 *'(3a = 0
-5 3'); ,-3a2) ' 2) 3'-/53) '('5 6
5.27
5.26*-'(-,1 -7 52 * -) :6) '3 ., .-/:1 (:-, * , 227 . (,3 ., . ((1 (:-, '),
a = 1-52 -(8'3 . (,3 ., ;,1 * , ) '3
(2) .) (' 3+-+1 1 27 -,'4, 3:.)1 2) -',:-2 , (3) -21-0;(, - (, -',:-2 (') 2) 31 (+2 3 (()3Y
) ('3 . (,3 ., (:4'; (2-,7 5()/2 ');, ,7 (:'.;) 3-53 2 6( 2.10)
-21 -0;(,3 *+413(( 2.7)
3,' 6, 1 2) (') 2) ,-3 3-53 9, ( +(/2 ; 27 '(5 (:'.; 3-82'(4 -, '+.3 -; -5) ((-7 6'+. -;24;3 . (;-;8 25
n(t))' ((.1 (-2, 6
5.31'(-,5 (. :7 4;3 . (;-;8 25 -,'4, -23.
X(t)3-3- 31 (+
f0'+. 5-5 0' '8 -23. 6 '(-,
3:(. : Y (t) = X(t) + n(t)
(.1X(t)
'(9/)2Hopt(f)
.-21-0;(,3 .::13 6X(t)
5 -(2. -.25 Sn(f) = 1/96
5.32'(-,5 (. :7 .-,':(
5.36 -
) - (+1 6.1--4 3:-,X(t)
.(,3 2) '+.3 .2(7. *35 *-'+. *.(,5 . (. (, 3'-51 3:-,Hopt(f)
) 52 * -)66 -5 31 (+ 3,' *-'/,3 *-'+.5 6 2) 3./:3
-21 -0;(, :1 6 '(-,-+- 2 *-:(. : ) '3 2) ( . (,3 2) .-2'04;3 . (;-;83
5.9
SX(f).=
S0
1 + (f/f0)2
Sn(f).= N0 .
( 5.36)
-+- 2 (. : -21-0;(,3 :13 -9,Hopt(f) =
SX(f)
SX(f) + Sn(f)=
S0
1+(f/f0)2
S0
1+(f/f0)2+N0
=S0
S0 +N0
(
1 +(
ff0
)2)
6 =
S0
S0 +N0
S0 +N0
S0 +N0
(
1 +(
ff0
)2) =
S0
S0 +N0
1
1 +
(
N0
S0+N0
(
ff0
)2)
6 =
S0
S0 +N0
1
1 +(
ff0γ
)2 , γ =
√
1 +S0
N0
6 * (7-2( S0
S0+N0→ 1
(:5 (γ → ∞ -7 254 :
S0/N0 → ∞ '1 (27 32(+ .(,3 2) .-/-3 3183 '),7 (:5 (
γ → 1-7 254:
S0/N0 → 0'1 (27 3:04 . (,3 2) .-/-3 3183 '),7 .,9 .1 (2 6
Hopt → 1,-3 (9 31 (+5 3,-)3 -7 .(: 31 (+ 5() -/ 6Hopt → 0
* (7-2( S0
S0+N0→ 0
E[ε2(t)] =
∫ ∞
−∞
SX(f)Sn(f)
SX(f) + Sn(f)df
6 =
S0N0
S0 +N0
∫ ∞
−∞
1
1 + (f/f0γ)2 df
6 =πS0N0f0γ
S0 +N0=πS0f0γ
6
) 7 (E[n(t)] = E[X(t)] = 0
/-:32 -)1: 6. -21 -0;(,3 .::13 .,-81 . -5 ., .8415 5-/'32 .-:'),7Y (t)
., -(8' . (,7 );/:X(t)
3-3- -(8'3 . (,3) *(415) 75 3-3. 35/'33 63 -82'(4 -'/X(t1), n(t2)
Y (t) = dX(t)/dt31 (+2 6
X(t),-3 3 -:73 '),7
d(t)*232 35(. * .-',:-2 .7'1 2) 3,-8-3 , (3
Y (t)3'413 ., 254:D(f) = Fd(t) = 1
'),7 65.33
'(-,5 '+.3 5/'15 .', (.1 .7'13 6'9( , (3d(t)
'),7
:(1 ( +5(1 35(. .(, 6 '(-,72(
5.34'(-,5
5.33'(-, ., -2/32 27 (: 3 -8-9(;';( - 6* +(43
:(1 ( +5(1 35(. .(, 6 '(-,
E
[
ε2(t)]
=
∫ ∞
−∞
Sn(f)|H(f)|2 df +
∫ ∞
−∞
SX(f)|D(f)−H(f)|2 df
3,53 3'(85 * ()': 2-2 3, ()13 .,E
[
ε2(t)]
=
∫ ∞
−∞
∣
∣
∣
∣
H(f)
D(f)
∣
∣
∣
∣
2
Sn(f)|D(f)|2 df +
∫ ∞
−∞
SX(f)|D(f)|2∣
∣
∣
∣
1− H(f)
D(f)
∣
∣
∣
∣
2
df
(:2 ) - -9, D(f)
5 ('.;3 ., 2-;7: 7 /, ( -21-0;(,(H(f)/D(f))
);/: -21 -0;(,H(f)
);/2 * (415 *,
72( 6D(f) = 1
3'415 (17 3-5 3. (, 4 (-+5(
H(f)
D(f)
)
opt
=SX(f)|D(f)|2
|D(f)|2[SX(f) + Sn(f)]=
SX(f)
SX(f) + Sn(f)
72(Hopt(f) =
D(f)SX(f)
SX(f) + Sn(f)
6 ,-3 .254.13 .-21-:-13 .8(113 .- (5-'3 3,-)3(
E
[
ε2min(t)]
=
∫ ∞
−∞
|D(f)|2SX(f)Sn(f)
SX(f) + Sn(f)df
372)33 ('4
2) 3'413 22(7 -21-0;(,3 :13 ., '84 ( '-) - (5)/5 5)/: -21-0;(, -',:-2 (:- 2 39 - *(7-2'(797 6
5.35'(-,5 193 '() -15 .', (.1 .7'13 6372)33 ('4 .'95 )'2 . (,3 -5 3-82'(4
193 '() -15 .7'1 ) (' . (, (:- 6 '(-, 6-9,
D(f)2) 37 (;33 3-' (; .'1.3 .,
d(t)5 1 : 6
r(t) = n(t) +X(t)
E
[
ε2(t)]
= E
[∫ ∞
−∞
X(t− θ) d(θ) dθ −∫ ∞
∞
r(t − θ)h(θ) dθ]2
.
*--42 .5--/ .-21-0;(,3 .::13) 254: 372)33 ('4 -E
[(∫ ∞
−∞
X(t− θ) d(θ) dθ −∫ ∞
−∞
r(t− θ)hopt(θ) dθ
)
r(η)
]
= 0
72( −∞ < η <∞ η27 '(5
∫ ∞
−∞
Rr,X(t− θ − η) d(θ) dθ =
∫ ∞
−∞
Rr(t− θ − η)hopt(θ) dθ
254: 3, ()13 -;, -:) 2 3-' (; .'1.3 85:(τ = t− η -8:
Sr,X(f)D(f) = Sr(f)Hopt(f)
72(Hopt(f) =
D(f)Sr,X(f)
Sr(f)6.()15 ' 1 , .r,X
(5 -2273 3'413 ., (:'.; 3) 12( r = X+n
')45 (:)1.)3 ,2 -7 52 * -) 3'37 (
Sr,X(f) = SX(f)*--4.1 3-82'(4 -'/
X(t)(n(t)
.(. (,3 (r = X + n
(5 -0';3 3'415 3'36'+.3 '() -15 / (. -: -+- 2 (:)3)( 5.43)
3,8(.3 ., *-2541(Sr(f) = SX(f) + Sn(f)6193 * (/.5 *-8(2-, (5)/5 ./42 ');,1 , (3 (:5 ( '. (- -227 , (3 372)33 ('4 -+- 2 /(.-:3 3'(,72
1 2(+ '53 * '5 1 *) --2 .-: ,2 2)12 '+.3 * (/.5 *-8(2-, *-1--4 . (-+:3 . (-55 *-.-2 *2(,6'+.3 '() -15 / (. -:3 '. (- / (: (2,7 . (-55 6'+.3 .5(. '+ '+(1 '533 '),7 (. :* +-+5 195 * *-1-) - -8' 195 * -7-23. '(5 39 4';5 (:/.-;) . (0 -)3( / (. -:3 -7 ) -+32 5 ()/ *(-26-/7 (:3 '(431 42/ *:-, * -0';3 , *-24 *--(:-) ';1
'1 (, * -2(+3 *-';13 4 (/ 39 3'415 6339 (2-; -25( * --(2. -.25 , 1 2) 3'+ X1(ω), X2(ω), . . . 3-3.*--4.1E |f(X1)| <∞ 3'(5)
f(x),−∞ < x <∞ 3-84:(; 27 '(5)1
N
N∑
i=1
f(
Xi(ω))
−→N→∞
E
[
f(X1)] 6
6. :7.1 *--,'4, *-:.)1 2) 3'+) *-'1 (, '),7 *-:-51 (:/:, 31 '-+32 '(8 ) - 5 (1 (9 3,8(.2 ..2 -+7563,53 3'(85 .,9 ,05:6339 (2-; -25( . 5 ;, .2/ (. -25
Yi(ω)3 -9,
Xi(ω) = E[Xi(ω)] + Yi(ω)*()':(
E[|X1(ω)|2] <∞) /-::3:0
E
(
1
N
N∑
1
Xi(ω)− E[X1]
)2
−→N→∞
0
2 8 3/7 (3E
(
1
N
N∑
1
Yi
)2
−→N→∞
0
*2(,E
(
1
N
N∑
1
Yi
)2
=1
N2
N∑
1
E[Y 2i ] =
1
NE[Y 2
1 ]→ 0
5 8-5 8 (-(() -, 3,' 5 (: * ,71P
1
N
N∑
i=1
|Xi − E[X1]| > δ
≤ E[Y 21 ]
Nδ2
6N →∞ '),7 2 2(3 -(0 -53 72(
254: E[|f(X1)|2] <∞ ) ('+:(
f(Xi)5 2;0 :
Xi5 2;02 * (415 *,) 52 * -)
1
N
N∑
i=1
(
E[f(Xi)]− E[f(X1)])
→ 0
65.44
2 -');, ) ('; 3((31 .,9 3,8(.6. -,'4,3 3'+5 *-:()3 *-'5-,3 -5 . (2. ) - '),7 (7 : ',) :
5.44 *, 32,)3 .)45.1
f31(/ 3-84 :(; 27
K27 '(5 *, -+(', ,'4: -23.3 6-':(-80 , . X(t),−∞ < t <∞ 3-3- 3'+3
. ('5.35 *--4.1t1, t2, . . . , tk
3'+ 27 ( * -:.)1K
2)1
2T
∫ T
−T
f(
X(t1 + s, ω), . . . , X(tk + s, ω))
ds −→T→∞
E f(
X(t1), . . . , X(tk))
0;)1 -1 - +85 ((:13 -,'4,3 3:.)12 2,1) ,5) , 13 *- :7.1 5(1 39-,5 32,)31 *-12.1 (:, 5 ()3() 7, 2(53 -.1 '1(, ,2 0;)13 25, 2(5 7, )- 2,1) ,2) '1(, -+(',3 0;)13 3)4 ( (+-251 :,3 8(112 3 () 193 8(11 *--+ (', *-7-23. '(5 31 -3 - ,05: .-+(',3 3:(7.3 ., 6.2/(.26.2/(.3 *--+(', ,2 * -7-23.2 . (,1 (+Xt(ω) = C(ω)
-,'4, 3:.)1 6((:1 ,2 , 1
A(ω)*, 6
[0, 2π]* (/.5 +-/, 2(;1 (
A5 -(2. -.25
φ'),7
Xt(ω) = A(ω) cos(2πf0t+ φ(ω))6-9,
1
2T
∫ T
−T
X2t dt→
A2(ω)
26= E
[
A2(ω)
2
]
6-+ (', 7, -23.3A(ω) ≡ const
'(5) . (,'32 ');,(-3- 6
Xt(ω) = cos(2πf0t+ φ)
Yt(ω) = cos(2πf0t+ ψ)
6[0, 2π]
* (/.5 +-/, *-2(;1 ( .5ψ, φ
'),7
'-+:Zt(ω) , Xt(ω)Yt(ω) 6-+ (', , . (::-,
Zt(ω)) . (,'2 24 :
6. -+ (', . (-32 .5--/ ,2 -+ (', +/, 27) *-:() * -7-23. -:) 2) .5('. 62) 195 . (/.;.33 2315) ) ('+2 -'8 -+(', 3-3- , .) .:1 2) *-:((:1 * -'412 -2 (, 0'; '('5 3) '352 27. : *, -9,
5.36'(-,5 ',(.17 . (. (, *-++1: *, (:--3+ 6. (-');,3 23 . (' (8 27 ., 254 - , (3 * + 27
* +1 . (-84:(; -.) 6 '(-,6(0, T )
045X(t, ω2)
-23.2 +(,1 31(+ -23.3 (-2)(t0, t0 + T )
,81: (', 4 -;1 19 '(5ω1
* * +3(',3 .--5 2) +,1 -42/ ('.;5 72 4;. : 63)4 3-5 ,-3 -+ (', , (3 (. : , . *, (542 3-53 '(1,7
.(-+*, 8(113 -52 -+ (', ,'4-- 5/'3 5(15 -':(-80 (2-;, (, -':(-80 X(t),−∞ < t <∞ , . 3'+3
E
(
1
2T
∫ T
−T
X(t) dt− µ)2
−→T→∞
0
*, (:--3+ E[X(t)] = µ
'),7E
(
1
2T
∫ T
−T
(
X(t)− µ)
dt
)2
−→T→∞
0 .
:,-' ((43 *, 0;)1KX(t1, t2) = E
[
(X(t1)− µ)(X(t2)− µ)]
*--411
4T 2
∫ T
−T
∫ T
−T
KX(t1, t2)dt1dt2 −→T→∞
0
68(113 -52 -+(', -23.3 -9,6. -+-1 3/7 (3
-':(-80X(t)
*, )135 3'3 3,' 3/7 (33 ., ,7 .-: ,2 3,53 3,8(.3 ., 2542 27 (: 39 0;)1 (.1.1--41ψ(τ)
*,(KX(t1, t1 + τ) = ψ(τ)
( 5/'3 5(151
2T
∫ 2T
−2T
ψ(τ)
(
1− |τ |2T
)
dτ −→T→∞
0 6
')43 2 .5.1 3/7 (33 68(113 -5 2 -+(', -23.3 -9,∫ T
−T
∫ T
−T
Q(t1 − t2)dt1dt2 =
∫ 2T
−2T
Q(θ)(2T − |θ|)dθ
65.19
'(-, 5-5 3-(81 39 ')42 3/7 (33 3'3* '-+: 8(113 -5 2 . (-+(', 2) 3'+33 /('5
1 : 6-':(-80 , .X(t)
3-3- 3'+3zλ(t) = X(t+ λ)X(t)3-3-
X(t)) .:1 2 72 63 -82'(43 -5 2 -+(',
X(t)) +-: -9, 8(113 -5 2 -+ (',
zλ(t), λ27 '(5 *,*--4.32 -'8 3-82'(43 -5 2 -+(',
E
(
1
2T
∫ T
−T
X(t+ τ)X(t)dt −RX(τ)
)2
−→T→∞
0
zλ(t)
2) 3-82-' (4 (0 (,3 .-84:(; .,Rz(τ)
5 1 :Rz(τ) = E
[
X(t+ τ + λ)X(t+ τ)X(t+ λ)X(t)]
-9,ψ(τ) = Rz(τ) −R2
X(λ) 62-2 5.45
-, :.3 ., *--42 -'8'+1 *-0 :1 (13 ., * ,2,
X(t)2) -:) '+1 *-0 :1 (13 ., 4' ,2 .+2 (:-2
ψ(τ)., .+2 . :1 2 3'3
;, .2/ (. * - (, X(t)
'),7 *--4.1 *--:)3 -5 ')4 ) - - (, 3 3'415 6- -5'ψ(τ) = RX(λ+ τ)RX(λ− τ) +R2
X(τ)63 -82-' (43 -5 2 -+(', -23.3 -9, '31 4-;1 ;,2 +'(-RX(τ)
*, 39 3'415 72(. (+-+1 -52 . (-+(',3 . (1)1
-,:.5 *3-2, * -'+()13 . (. (,2 *-02413 .5(. ., 4 (+52 . :1 2 6. (-(175 '8(-13 0241 (7 .7'1 3:(. :. (, '(0':1 57'(1 '(413 6* -'+()13 . (. (,3 ., '8--13 '(41 32-713 3+-+1 .7'1 *-:(5 (:, -:(8-/ )'
.(+-+13 (.1 27 (:( .7'13 . (3:.3 ., +(+1:( * -'8(-13 *-02413 +/, ., '5/: (9 .7'12 6)' '(0':(2) 2(+ ';1 -:; 2 3. (,: 3'(85 81:( . (+-+13 2 '(9/: *, 60241 (. (, 2) (5-0 '(5 .(,8(. 25422(+ ';1 2 *, -5 ( ++(5 0241 2 *, -5 . (+-+1 .(, 275 6* -0241 ( (. (, 5-0 2 37'3 254: * -0241) ' -' (0': ';1 2 0241 27 4 (+52 * -7-'8 (:--3 -2(, *85 6) ' '(0': (. (, .(,-815 ',): *-0241 2)');, 72( -+ (', , (3 ) '3 '(0':1 '8(:3 ) '3) *-/-:1 (:,) ,-3 35().3 (+1 6.,9 * -) ( ,2 *2(, 812(5--/ ++(5 * + 27 -+ (', 3-3- -23.) .:1 2 3) '35 '57 '5 (3) -;7 ( . (-3 +/, )' '(0':5 4;.326'+()13 . (,3 -5 2 * ;(. 2(4 -) (. (, 62542 2(7 - -23.3) . (-');,3 23 . (' (8 27 ., 2542 .('5.35 .(,) 3/:35 5 () 6812( . (, -' (0': 35'32 0241 27 '5/2 *-7-'8 (:--3 -,'4, . (, , (3 '+()13 '(413 *,4;.32 27 (: -+(', , (3 -,'4,3 '(413 . (,) 6+-/- . (, '(0':5 4;.32 27 (: -+(', ,(3 -,'4,3 '(4136+-/- . (, '(0':5
(2) .-2'04;3 . (;-;83) ;, .2/ (. 25 )' '(41 *--4 ,53 3'415 . (5' *-4 ( .(-) 1 . (-55'-(817 0' 5/' '(41 +(,1 *-3 (5 *-'+.2 + .7)1:( +(,1 * -7 (1: *-'+.5 32-/.1
0' 5/' '(41 6 '(-,04 (2) 0'3 5/(' *2(, 0' 5/' -2(, '5 13 6
K/.1 .'53 (
∆F0' 5/ (' 25 '51 '+ '5 (1 39 )'3) 12 .-2'04;3 . (;-;83 '517 2 (; '513 (5) *-'+.3 * (/.5 .('/, *-2-15 6) '3 2) 391 35'356'5 131 3,-8-5 .-5-04;,3 ) '3 .18( 31 ,-3 32,)3 63 (54
) ' 2 '5 1 .;)3 6 '(-,,-3 3,-8-5 . -5-04;,3 ) '3 .18(
√
Ropt(0) ∼=√
K2N02∆F
3-53 '(5 5'(41 2+(1 * -:(5 (:--3 (. :3Sn(f)
* (415 *, 6N0
39 5 ()/) 31 63,-8-5 3-;(1 ,2fc
) 52 * -).-2'04;3 . (;-;83) ) ' '(41 6Ropt(0)
'(5 3,8(. 3. (, * -2541 (:--3 *-'+.3 275Sn(f) ≡ N0
*-/-:1 (652 )' ,'4: * -'+.3 275N0
,-3 (2)*,) 52 * -)
Sn(f) = N0
-9,Rn(τ) = N0δ(τ)6∫∞
−∞N0df =∞ - (. : 4;33 -7
Sn(f) = N05 '57 3 -; (1 (9 3-5 6
Rn(0) ' ∞ 9, (6-; ( 8(11 4;3 * -23.7 *--4 (:-, , (3 -24 -9-; -,'4, -23. 2) 3-89-2, -+-, , (3 52 ) ' .('/, *-215Sn(f)
2) *-7'3 *- (+- ,2 . (5('4 *-.-2 . (,-815 631 (+5 (:',.) -;7 -24 -9-; -23. 2) 5('47 (. (,'2 ) -(:. (, * -:--:13 *-'+.5) , (3 (+-) 27 6*--(1 '+.2 21 , (3) (+- 4' (+- (:-,fc
(2-;, ( * -3 (5 *-'+.53-84:(;2 .-; ( 3-':, /-2 ');, *,3 -0 :-1'0+3 3'415 5 , N0
31'5 3 (54 .-2'04;3 . (;-;8352 )' '(41 '(5 6+ (,1 / (: 5 ('43 32, *-'415 72 6
δ(t)
3,-8-3 4;3
∫ ∞
−∞
N0|H(f)|2df =
∫ ∞
0
2N0|H(f)|2df
.-++8 +/ . -2'04; .(;-;8 *-'-+13 ) --++8 +/
S(f) =
0, f < 0
2S(f), f > 0 9, (3,-8-3 4;3
-++8 +/N0 ·∆f ·K2.-+:3 3+(55 (2-, ( .-++8 (+ . -2'04; .(;-;8 (:'+3) -;7
S(f)., '-+32 /(: * --0'(, -. . (:(5)/5*3 3'+3 (9-,5 ,+(2 -'8 '1,1 (, ';5 (- 2) 3'41 275 6. -++8 +/ .-2'04; .(;-;85 )1.)32 / (:6. -++8 (+3 3'+35 )1.)32 -)1: (:/:, 6* -)1.)1
142+7 52 ) ' - '8(: (2-,7 52 ,2 ) ' 27 . (,'2 ');,
5.7(1 * ('04; * )' '(8-- 6 '(-,. (-(');,3 ./,5 '(/52 ');, -9,
S(f) = 11+(2πf)2
*, 2)12
H(f) =
1
1 + i2πf
1
1− i2πf
1√
1 + (2πf)2
.(,'32 ');, 6(9-, .-.5- ,-3 ) (2)31 ./, 4') 52 * -) 6S(f) = 1
1+(2πf)2., .-. 31 ./, 27 (*,)
Paley-Wiener2) (-'0 -'43
∫ ∞
−∞
| logS(f)|1 + f2
df <∞
652 ) ' 3 -:75 '),7S(f)
.-2'04; . (;-;8 3,-8-5 .:. (:3 .-.5-H(f)
.1--4 -9,'9(5 --:
) ' 2) 3'-9 6 '(-,S(f)
'(5 *-,53 *-'415 --:
S(f) =
N0
1
1 + f2
1
1 + f4
'(92 ');, 32, . (' (411 (2-,6;(. (::-, 52 ) ' 2) 5('43) -'3 '9(5 '5 (+1 *,) ,(3 27)33 ' (1
52 )' 2) *-2'0:-,. (-0 -:-1'0+ . (-84 :(;
h1(t), h2(t), . . .3:-3. (
N0.-2'04; .(;-;8 * ) ' n(t),−∞ < t < ∞ 3-3 -6∫∞
−∞h2
i (t)dt <∞, i = 1, 2, . . ..-;( 3-':, . (25
1 :Xi =
∫ ∞
−∞
n(t)hi(t)dt
+,1 2(+ 0' 5/' 25 -23. n(t)
52 ) ' 2) 3'+33 /('5 -9,E[Xi] = 0
E[Xi, Xj ] = E
[∫ ∞
−∞
n(t)hi(t)dt ·∫ ∞
−∞
n(s)hj(s)ds
]
=
∞∫∫
−∞
hi(t)(t)hj(s)E[
n(t)n(s)]
dt ds
= N0
∫∫
−∞
hi(t)hj(s)δ(t − s)dt ds
= N0
∫ ∞
−∞
hi(t)hj(t)dt
3-82'(4 .8-'01 ( ;, .2/ (. * - (, -,'4, '(04 ((X1, X2)
- (, 52 ) 'n(·) *, +/ (-15 (
E[X21 ] = N0
∫ ∞
−∞
h21(s)ds
E[X22 ] = N0
∫ ∞
−∞
h22(s)ds
E[X1X2] = N0
∫ ∞
−∞
h1(s)h2(s)ds
*--(2. -.25 , 1 (-3-X1, X2
) .:1 2 39 3'415h2(·)2 h1(·) -5 ')43 . (-32 -'8 31
*-'+.5 ;,1 3:() ( *-3 (5 *-'+.5 ;, 017H(f)
(:--3+Low-Pass
(1H(f)
.'(1. .-84:(; 3:(. :66.20
'(-,5 ',(.17 *-7 (1:
. -) 1 *-7 (1: 3'-51 .'(1. .-84:(; 6 '(-,
3,53 3'+33 ,-3 .25 (41 3'+3 6(97 .'(1. .-84 :(;2 -5-04;, 0' 5/ (' '-+32 * -8('∆F ,
∫ ∞
−∞
|H(f)|2df
2|H(0)|2
-52N0 = 1
* 52 ) ' 3 -:75 '),7 3,-8-3 4;3 -5) /-3 , (3 ∆F
) '2 -5-04;,3 0'3 5/(' '1 (276. ('/, . ('+3 * )-) '(792 -,+7 , . (-5 35'32 -+12 3/ (: (9 3'+3 62|H(0)|2
,1 (+H(f) =
1
1 + i ff0
72(H(0) = 1
63 db
2) 0'3 5/(' , (3f0
-9,∆F =
1
2
∫ ∞
−∞
|H(f)|2df =f02
∫ ∞
−∞
df/f01 + (f/f0)2
=π
2f0
6π2 f0
, (3 ) '2 -5-04;,3 0'3 5/(' 72('/, '+.2 /-.32 * ');,
H(0)2 -5-04;,3 0'3 5/(' ., '-+32 * (415
∆F =
∫ ∞
−∞
|H(f)|2df
2|H(f0)|2.'(1. .(-84:(;2 (:--3+ Band Pass
(;01 *-'5 1 5 (17 22(7 .'(1. .(-84 :(;2 +/ (-15 3/ (: (9 3'+3 (;-031
0' '-51 :1 6 '(-,
(Nyquist
)
+:3) .:1 2 +:5 . (-32 *-5--/ 32, *-:('042, 6. -,'4, 3 (:.5 * -,81:3 *-:('042, *:) - -24-9-; +: 275*2(, ;, *32) 8(113) +:3 -:; 2 *-/.1 .'8(- +:5 .-,'4,3 *-:('042,3 . (:. 6*'9 '-532 7, 27 (-) ' '(41 ('85 -2,-+-, +:1 57'(17 -24 -9-; +: 27 . (,'2 (:-2 72 6;, (::-,) ) ' /.1 , (3 -'3 /.13
) (' +:2 2+(1 6 '(-,., , (812 ,-3 3-53( -':(-80 -,'4, -23. , (3 +:3 ) ') /-:32 27 (: 35-53 * 24)1 -((-) -, :.5* -'1 (/3 2)
T3'(0';103 2)
R.22(73 . (+:.33 2) '+.3 2) 3-84:(; -2(, 3-3-
Sn(f)-227 ;(,5 6
Sn(f).('(41 ,22 *-254 ( *-2-2 *-+: .)' (, +:5 * -4 ( (:,) ) -+: 6(2) 3-'01 (,-3 2) ( +:3 57'(1 *31*-2-2 *-254 *-+;3 -'/ *-0 :12, * *,3 3,53 32,)3 ., 2,) : -)1:) -:;2 6* --:(8-/ 4;3
,-3 35().3 +: (17 *32) *-4+33 2 )' * -'8(- * --2,-+-, *-'(01'(; :'0
6) ' .'8(- 3::-, * -+;3 .'/ .)' , 0;)1+:2 .'5(/13 *-+;3 .'/ .)'5 --: 3/7 (3
* -+;3 .'/ .)' 6 '(-,(2, * -/.1 ) ' -/.1 .'8(- 3.-3 *-+;33 .'/ .)'3 *, 6-1:-+(1'. 24)1 -((-)5 .,81: .7'13) /-::(.)'( '/,1 *-+;33 .'/ .)'3 ., *1/2 *-2(7 - * :-, +:3 2) ) '3 -/.1 19 (. (,5 6+:3 ., *-11/1 (-3+(-:5 .,9( .''4.1 3.-3 .)'3 (2-, ( *1/.1 3-3 +:3 3,8(.3 64;3 2542 32(7- 3::-, * -+;3 .'/6)' '8--2 32(7 - 3::-, *-+;3 .'/ .)' 72( -');, -.25 397 581 72 634-1-:-+(1'.3 2) -:)3 4 (/2
+:3 * -' (03 ) '3 '(41 '(5 5 0;)1Sn(f) = 2k TR
.-) 13 3/ (:3 3'3 61.38 ·1023 Joul/oK
182(5 2) 30:0 :(43 ,-3k'),7 +(,1 . (3 (5 . (-('-+.2 +6
∆f*-'+.3 * (/.5 (5-'5 /.13 8(11 = 4kTR∆f
) (7 : +-1. *(41 271 Sn(f) = 4kTR
,-3) (' +: -, (9 3,8(. -;2 6+:3 2) 3-'01 (,-5 (, * -'1 (/5 3-(2. 3::-, ( . -2'5-:(, ,-3 3,8(.3) 52 * -)3,8(.) )-+32 ) - 6* -'8(- *3) )'3 2) 0513 .+(4 :1 3+-1 3. (,5 * -' (, * -5(0 * -+:3 27 04) +:(* -1'9 (7'+ *-1-'91) +:5 '5 (+1)7 3:(7: 3::-, ,-3 *2(, .++(51 .7'15 '5 (+1 '),7 3:(7: 02/35 (96, -5:) 39 0;)12 .-42/3 3/7 (35 '35.. (9 3+(4: 6* --:(8-/'),7 . (::1 ( * - :+;1-, * (,. -21 . (81,5 39 2, 39 *. (, '5/:
Rb(Ra
*-+: -:)5 -- : .-42/ 3/7 (3* ( 318 -52 3:-5 -1 :-+(1'. 24)1 -((-)5 .,81: .7'13) /-::( *-+;3 -'/ . (::13 ( * (,.3 -2135-53
+: ) ' 5() -/2 21 6 '(-,
142+7 '. (- 30 (); 32(4) .7'1 00') :
+: ) ' 5() -/2 0) (;1 21 6 '(-,(:--3+ 0' .'8 .::1 '+ '51 -'/, +:3 2) *--'3 *-) '3 *3
nb(t), na(t)'),7
E[
n2a(t)
]
≈ San(f0) · 2∆f
E[
n2b(t)]
≈ Sbn(f0) · 2∆f
6nb(t)/2Rb
, (3 -:1 -3 '(4131(na(t)/2Ra
, (3 -2,1)3 '(4131 ,8(-) *'93 *- :+;1-, * (,. 2) 3/:35*- :+;1-, * (,. *--4) '(79: ,(3 -:1-3 +:3 1(2 ' (1 -2,1)3 '(413) 4;33
1
2na(t)
na(t)
2Ra
,(3 -2,1)3 1(2 ' (1 -:1-3 '(413) 4;33 31(+ 3'(85n2
b(t)
4Rb*-4;3 2) 8(11 (9-, * --4.32 5--/ -1:-+ (1'. 24)1 -((-) 2) 2(4 -) (.1 72(E
[
n2a(t)
4Ra
]
= E
[
n2b(t)
4Rb
]
(,Sa
n(f0)
Ra=Sb
n(f0)
Rb
.(-32 2(7 - (::-, 25,T
3'(0';105 (,f'+.5 -(2. . (-32 2(7 -) -2'5-:(, /- , (3
Sn(f)/R/-3 3:4130 (); 3-'01 (, - * +: 2) 39-2:, -) 3/7(3 ,22 254: 63-'01(, -5( '1 (/5 -(2. ,2 6, 69 +:3 (5 -(2.. (,'32 ');,
Sn(f) = 2k TR
4-;1 *-'+.5 3:(7: (93 3,8(.3 . ('-+. (9-, + 6* -+:3 27 '(5 .1--4.1 3,8(.3 (:,53) 2(4 -)3 225(3,8(.3 - -5 ;(,5 6+: . (-32 4-;1 +:3 ,2-11 23 '(, 2) 2+( '+5 *3 +:3 -+1-1 '),7 *-3 (56. (-0;(, . (-('-+.2 5 ('4 + 3:(7:.('3
(:--3+ *-4;3 2) -5 2 *-17.1 ) '3 -/.1) 52 * -) 6,
2(4) 21 '(05 *-+: ' (5-/ 6 '(-,* -7'3 ., (254 - ) '3 4;3 3( 32(4)3 . (+:.33 3) 7 * -+:3 -:) 2) .-5-04;, 3'(0';10 '-+: 72((,
2k(T1R1 + T2R2) = 2kTeff(R1 +R2)'1 (27 *-:(7:3
Teff =T1R1 + T2R2
R1 +R2
-' (0 3'(1. 21 * (415 -2-541 3'(1. 21 65
+: ) '2 -2-541 3'(1. 21 6 '(-,6-2-541 *'9 ' (417 ) '3 * 3'(1. 215 -2/32 (+-7 ');, -' (0 /.1 '(417 )'3 * 3'(1. 21., 3(() :
Si(f)*'93 2) 39 *
Sv(f)/.13 '(41 2) .-2'04;3 . (;-;83 -5 ')43 ., , (812 .:1 2
i(t)R = v(t)*-4+33 -:; 2 *4-' /.13
72(R2E
[
i(t+ τ)i(t)]
= E[
v(t+ τ)v(t)]
,71 (Si(f) =
Sv(f)
R2= 2kt TG6
G = R−1 '),7Si(f) = 4k TG
254:Sv(f) = 4k TR
'(5 ( .-5-; .)'5 --: 0;)12 '(5:
21 2) ) ' 6 '(-, 6 '(-8
.)'3 2) 3 -:73 -4+3 -:; 2 ) '3 31 63.5-5 * -1 :-+(1'. 24)1 -((-)5 .,81: .)'3*2(, *-4+33 -:; 2 )'3 7 3 ., );/2( (2) ) '3 '(41 ., +: 272 - (32 5(17 27 (:
, (3 -1:-+(1'. 24)1 -((-)5 .,81:3 . -5-; .)' 2) 3 -:73 :+;1-, *, 0;)1Zin(f) = Rin(f) + iXin(f)
3-3- 213 -4+3 -:; 2 ) '3 /.1 2) .-2'04;3 . (;-;83 -9,S(f) = 2kT Rin(f)
v2 =∫∞
∞2kTRin(f)df
3-3 - ) '3 4;3 7 3 (2 (4 -)3 6* - :+;1-, * (,. * (
f02 5-51 0' .'8 .::1 . (81,5
R+: * (. :3 213 ., '5/: 3/7 (36.)4 (513 3,8(.3 ., +-1 (:2 .(: 254.13 4;32 3 () '1:3 4;33 -1 :-+(1'. 24)1 -((-) 2)
215 --: ,1 (+3'(1.3 21 - *-4+33 -:; 2 ) '3 ., 5)/2 27 (:
0 (); 215 ) ' 6 '(-,
0 (); 215 ) ' 3'(1. 21 6 '(-,254: 3,-8-3 -4+32 ) '3 /.1 '(411 .'(1.3 .-84 :(;
H(f)'),7(
S(f) = |H(f)|2 · 2kTR =
(
12πifC
R + 12πifC
)2
2kTR =2kTR
(2π)2f2R2C2 + 1 (:254) 0;)13 -;2Zin(f) =
R · 1i2πfC
R+ 1i2πfC
=R(1− i2πfRC)
1 + (2πfRC)2
72(Rin(f) =
R
1 + (2πfRC)2 63,8(. 3. (, 5 (17 .254.1(2-'.
.(0 -) -.)5 *-4+33 -:; 2 ) '3 ., ,81 6R
.(+:.3 25 '(30 +: (17 3:.1 213R =
√
LC
'(52) ) '3 2 32, . (' (41 .;)3 5() -/ ( * -+:31 +/, 272 )' '(41 .; (3 - 5 6 0;)15 ) (1) - , 6213 -4+3 -:;3,' +/, ((72 32 '1:3 4;33 27 ., .:'(4 ,-3 '(+-) .:0 :,7) .-:((-7 3:0 :,5 -- : 39 -2 * (- .'3
38(/3 '4 (1 32 '1:3 4;33 27 (:--3+ *-+;3 .'/ ,-3) ( '(-83
-2-541 -' (0 21 6 '(-,
.-:((-7 3:0 :, 6 '(-,*-,53 *-:((73 +/, 2, 3;(8 3:0 :,3) /-:: 6
Rin'(30 -13 (, , (3 3:0 :,2 3 -:73 :+;1-,) /-::
61000 MHz
55o K
72 +'(- 100 MHz
51000o K
2) 2+( '+5 52/3 2-5)2 58-:5 22/3 ,
61000 MHz
515o K
72 +'(- 100 MHz
54000o K
2) 2+( '+5 52/3 2-5) 5 300o K
7 &',3 '(+7
6-((24 . (21 -;2, 2) 2+( '+5 +525 )1)2 -'4.) 7 4 -;1 3'8 3:-'4 .1(2, /-:: )1)3 + 3:0 :,3 2) 3'(1.3 .1-7 3-3. ) ' 2) 0513 .+(4 :1
.-:((-7 3:0 :, 2) 2+(1 6 '(-,6. (1-,.13 . (' (0;103 ., -832 ) - * -'4131 +/, 275 '),7
(Shot noise)
*-)';3 * 3-:1 -23. , (3 (, (; -23. '(797 6(, (; -7-23.2 .; (. ,-5: 3+(-+3 ) '2 /-.:) -:;2-2() (2-; ( * --(2. -.25PNt = k =
(λt)k
k!e−λt
6-23.3 584 , (3λ
'),7(, (; (2-; -;2
K-,'4, ';1 2-': *-(1
T54: 142+7 5)/1 2 -,'4, -23. 2) 3-82(1- 85:3+-/, 3'(85 *-2(;1 ( * --(2. -.25
[0, T ]5 * -:1 9
K-,'4,5 '/5:
K., (:2'3) -'/, 6
λT.2/(. *.7
(t1, . . . , tK))+/1 *:1 :( 32( '+5 *-:1 93 ., '+ :
(θ1, . . . , θK)32,3 *-:193 ., 1 : 63 93 * (/.5
142+7 32( .('+1 . -84 :(; *31 3:5:( t1 ≤ t2 ≤ · · · ≤ tk
(, (; -23. 6 '(-,3:0 (, (; -23. 2) . ('5.33 4 (/ ,(3 (:,813) -,'4,3 -23.3 2) . ('5.33 4 (/ 3/7 (3 ,22 3:0K
) *-+(- (:, *, ( .1 (82 3 -1 .-:(71) -,'4,3 -23.3 27 (, (; -23. 2 5 ()/: *, +(,1 .-:(-3 (9(2-;) /-:32 '-5 ( .'/,3 2 .+(- ,2 .1 (82 (--) . (-:(7131 ./, 27
[0, T ]193 045 .1(82 ( -3 . (-:(716
(0, T )5 +-/, (2-; 3-3- .1(82 3.3
. (1)1 3+(-+3 '+ (54 I0
*'9 * -1 -'91 (:,) /-::( ('042,3 01 .,q5 1 : 3+(-+5 *'92 .7 '(5:63+ (:,2 3+(0431 *-'5 ( *-:('042,
I0/q3-:) 275) ,-3 '5+3
0' 5/' '(41 6 '(-, 6 '(-8
I0'(30
D.C.*'9 +:3 ( 3+(-+3 '+ '5( 3-3 3-:)5 * -:('042, ( :-, ( ;, 3-3 ('042,3 01 *,3.-3 *'93 2) .-2'04;3 . (;-;83 (
S(f) = I20δ(f)3'(831 .-2'04; . (;-;82 3;8: -; ( 01 -25 * -:('042, - *': *'93 ( '/,1
S(f) = I20δ(f) + Sn(f) 6
Sn(f)., 5)/2 ,-3 (:.'01
72(ie(t)
*'9 215 3')1t = 0
'5 02;:3 ('042, (:2) 2+(15('042,3 01
= q =
∫ ∞
−∞
ie(t)dt
/-:32 '-5 3+(045 *-:() * -' (9-,1 *-02;: * -:('042,3( '/,1 6* -:('042,I0T/q
8(115 (02;-T
193 045./:935 72 6(, (; -23. , (3 *-:('042,3 .0-2; -23.) /-:32 72 27 (: 630 -2;3 -:1 9 -5 . (2. -, ) -)2 (+
T6, 69 .((843 04;,
I(t) =
K∑
k=1
ie(t− tk)
72 6[0, T ]
*(/.5 +-/, *-2 (;1tk
(λ = I0T/q
(, (; (2-; -;2 2 (;13 , 1K
'),7E[
I(t)]
= E
[
E
[
K∑
k=1
ie(t− tk)
∣
∣
∣
∣
∣
K
]]
= E
[
K1
T
∫ T
0
ie(t− θ)dθ]
=q
TE[K] =
q
T
I0qT = I0
X, 1 '(5
Eg(X) =∫
fX(θ)g(θ)dθ3+5(5 * -)1.)1 -:)3 (-(()5 342/33 .:(7.1 , (3 (),'3 (-(()36(, (; -23. 2) .2/ (.3 3 -; (1 - -5'3 (-(()5( . ((843 ./:93 -(0 -5 -+-2 3,5 -) -2)3 (-(()5
g3-84:(; (
-:)3 0:1 (12 .7 3:;:R(τ) = E
[
I(t+ τ)I(t)]
= E
K∑
k=1
ie(t+ τ − tk)
K∑
j=1
ie(t− tj)
= E
[
K∑
k=1
ie(t+ τ − tk)ie(t− tk)
]
+ E
∑∑
k 6=j
ie(t+ τ − tk)ie(t− tj)
04;, ./:935 72 6K
-:; 2 7 /,(tk
3 -:; 2 32-/. 81: 6[−T/2, t/2]
*(/.5 +-/, *-2(;1tk
3) .7 /-::K
2 (81 1 -E .((843
R(τ) = E
[
K∑
1
1
T
∫ T/2
−T/2
ie(t+ τ − θ)ie(t− θ)dθ]
+ E
∑∑
k 6=j
1
T 2
∫ T/2
−T/2
ie(t+ τ − θ)dθ∫ T/2
−T/2
ie(t− η)dη
∼= E[K]
T
∫ ∞
−∞
ie(τ + θ)ie(θ)d(θ) +E[K2 −K]
T 2q2
(,(; -23. '(5E[K2 −K] = (λT )2 ; E[K] = λT
72R(τ) =
I0q
∫ ∞
−∞
ie(t+ τ)ie(t)dt+ I203-' (; .'1.3 - 254:(
Ge(f) = Fie(t)5 1 :S(f) = I2
0δ(f) +I0qGe(f)G∗
e(f)
Ge(f) ∼= Ge(0)*--4. -
τf 1*-1--413
f*-'+. '(5 3+(:,2 3+(0431 ('042, 2) '513 19
T3-3-
72(Ge(0) = q
(:2) 3'415(S(f) ∼= I2
0 δ(f) + I0 · q (,Sn(f) = I0q 63+ (-+3 ) ' ./ (: ,-3 (:254) 3,8(.3
.('36q
('042,3 01 ., -:(- : ;(,5 ,(812 ');, 3+(-+3 )' .+-+1 - ( (:254) 3,8(.3 .'95) 52 * -) ,
(+12I0q
,/ (:3 *-,8(1 '9 -';5 5 04;, ,'4:3 (: 04;, -; (1 * -2(+ 3+(-+ -1'9 '(5 6-+1 *-2(+ ,2 3+(-+ -1'9 ' (5 ,-3 (:254) 3,8(.3 -5/'1 01 *-'8(- *3 3+(043 ., *-:('042, 35'3 *-59( *-(1 '5 *, 142+7 ,(3 ( -5/'13 013- (. :31 (1: 3+(-+ )'2 *'( 39 04;, 6* -; (: * -:('042, 2) 30-2;3 ., 5713 3+(:,2 3+(043 -5 6/ -:9 , (3 -+1 * -3 (5 ,2 3+(-+ -1'95 *2(, (:254) 3/ (:3
3+(-+5 -7 '-79:I = I0(e
q V/KT − 1)
7 3+(-+5 * -1'93 ., 2+12 .-: 3) 12(I0 e
q V/kT
−−−−−−→3;-/ *'9
−I0←−−−−−−3-9(;-+ *'9
72 6 .5 *--:, (; * -1'9 *3 3-9;-+3 *'9( 3;-/3 *'9Sn(I) = qI0(1 + eq V/kT )
V = 0
'(5 (Sn(I) = 2qI0
6 3+(-+3 '(5 -7 52 *-) : .-),' 213 ., .7 '5/:
R
∂I
∂V=qI0kT· eq V/kT
+: (17 3+(-+3 . 3:.1 0/.1 5-5 72(
∂I
∂V
∣
∣
∣
∣
V =0
=qI0kT
=1
R
6 72 6* - :+;1, * (,. *--4.1 397
R.'-/5 * 72
2qI0 = SR,In
,71(SR,I
n =2 kT
R
/.1 ) '2 '515 ,71(SR,V
n = 2kTR 6
6(., -8-5 (1: ) '( 37 (1: 3'53 * -:)3 ( (.,-8-5 3 (5 ) '( 33 (5 3'53 * +/,3 *-'5 1 -:) *-:(. :) /-::6-/7 (:3 -5 4 ( : 39 (1 . (-55 -:)3 -:;2 *-)2 -+ *3-:)1 39-, -:)3 '/,2 +/, *31 39-,4;3 '5 3 -(81 4;3
254.- 39 4;3 6' (4131 2542 ');,) -21-71 4;3 '(413 2) -(81 4;3 '-+:( * -(1 '+.5 --:-(813 4;33 254.-
6.21'(-, '(5 72( * - :+;1-, .1,.3 -,:.5
'8--1 , (3) ) '3 ( 21 6 '(-,- (. : -(813 )'3 4;3) 254:
∆f*-'+. * (/.5 -1'0 ) ' '(5
E
[
e2(t)
4R(f)
]
=2kTR(f)∆f · 2
4R(f)
= kT ∆f 6-5-; -24 -9-; '(415 -(813 ) '3 4;3 39(.('+3
-5 -04, (, -5-; , (3)27 '(41 2) .-5-04;,3 3'(0';103 6,Teff =
1
k ·∆f · [∆f0' 5/'5 -(813 )'3 4;3
]
3,-8-5 -(81 4;33 -:75 -(81 4;3 = Ga4;3 '53 65
.7'15 --: 6) ' .'; ( '51 2) .-5-04;,3 ) '3 .'(0';10 *-) (13 2 .'8(41 3'-4 ,-5: 2326' (413 .'(0';10 2) 3-84:(;7 3,-8-5 ) '3 4;3 ., '--8:((Tin)
'(413 2) 3'(0';103 ., 3:) : .'-(813'-(817 .'+(1
∆T.-5-04;,3 ) '3 .'(0';10 6'5131 42/ ,2 , (3
R0'(413 52 * -)
'5 13 ) ' 6 '(-,3-3- ,2 7 3) 7 '(413 .'(0';10 ., 32:( )' ,22 25, 339 '515 ) ('3 '513 -2/: . ('/, *-2-156
∆T35(). - (32 -'8 317 6-(:-)
3+-+1 '5 13 ) ' 6 '(-,
3-3-G
'5 3(∆f
0' 5/ ('5 '513 .,-8-5 -(813 )'3 4;3 3'+33 '(,2G · k · (Tin + ∆T ) ·∆f
-9, T 2 3;(8 3:0 :,3( * -+;3 .'/ 3:0 :, ( .'(1. (4 '+ '5 (/1 '5 13 *, 72630 -243 ., ';) -
∆T5 '(;-) -9,
T ∆T*,63) 12 ';) - ,2
∆T'(;-) -9,
∆T T *,
*-- (;-0∆T
-';1T0 = 300o K ∆T ∼= 10× T0
32(1 ,2 * -2 ('4 -1 0241−4× T0
2(1 *-2 ('4 -1 0241∆T ∼ 80oK
-'01'; '5 13+44 39 '/, 395 *-'5 1 -:) 2) .22(73 . -5-04;,3 3'(0';103 3:0
'(05 *-'51 '(5-/ 6 '(-,Gk∆f(T + ∆T )
, (3 ++(5 '51 2) 3,-8-5 )'3 2) -(813 4;33 (:'3 '57) -;7 .-),' 3/7 (3'51 .,-8-5 -(813 ) '3 4;3 3+44 '(5 6'(413 2) ) '3 .'(0';10T ( -(813 '533
G'),7
-(81 ) ' 4;3 3-3 -b'51 2) 3,-8-5 72(
Gak∆f(T + ∆Ta)- (. :
a
GbGak∆f(T + ∆Ta) +Gbk∆f∆f∆Tb = GaGbk∆f
(
T + ∆Ta +∆Tb
Ga
)
72(∆Tab = ∆Ta +
∆Tb
Ga6a-:;2
b., (,
b-:;2
a., *-)2 -+ *, 0-2/32 27 (: ,/ (:31) 52 *-)
35-53 .'(0';105 +: '513 . -:72 '5/: 142+7 .'+(13F
) '3 .'; ,(3 '513 )'2 (: -;,1.7 '-+: 6
300kG∆f5(17 3-3- ' (413 2) -(813 4;33 6
300oK
F =
'5 13 ) ' 22(7 '5 13 .,-8-5 -(813 ) '3 4;3300k∆fG -
∆T2 '()4
F72(
F = 1 +∆T
300) '3 .; (. - '513 ) ' ., --;,1∆T
) 52 *-) 6F = 1
(∆T = 0
) ' '8(- (:-,) -2,-+-, '51 '(56300oK
2) 3'(0';105 '(413 '),7 '(413 ) '2 2;7 *+41 - '513 ) ' ., --;,1F
(2-, ( '513 *'(),(3∆T
.-. ('-') (/ - .+(4 : .)'(+ 3:-,∆T
.'+3 (2-, ( . -. ('-') 3'(0';102 /-5 '+(1F
) (-716'. (- - -5 ) (13+445 *-'51 -:) '(5-/ '(5) /7(3 2-'.
Fab = Fa +Fb − 1
Ga
Matched Filter
1 : 6t > 0
'(5u(t) = 0
) /-:: . (0);3 12 6u(t)
.-;( 3-':, 25 -0 -:-1'0+ . (, 3 -:73 . (,6.25
'(-,5 -- 6U(f) = Fu(t)
) '( . (, 6 '(-,6N0
.-2'04; .(;-;8 ;, .2/ (. 52 )' 3 -:73 )'
) ' ,22 +525u(t)
. (,3 -(81 3 -:75 '),7 3,-8-3Y (t) =
∫ ∞
−∞
u(t− θ)h(θ)dθ
3-3.t = 0
'5 63 9 . (, 2 +-13 27 ., 22(70
195 ,8(13 ' *--5 (-/ * -:1 95 ;,.1 -(8'3 . (,3) ((-73,-8-3Y (0) =
∫ ∞
0
u(−θ)h(θ)dθ
142+7t = 0
'5 3,-8-5 ) '2 . (, /- '-+:(
S
N
)
outt=0
=(Y (0))23,-8-5 8(11 ) ' 4;3
'+.3 5/'15Y (0)
., *()'2 27 (: 2(('; 0;)1 -;2 6');,3 277 2(+ 3-3- 39 /-) 38':(Y (0) =
∫ ∞
−∞
H(f)U∗(f)df
8(11 4;3 3,-8-5 ) '3 4;3∫ ∞
−∞
|H(f)|2 ·N0df
5(: 2(('; 0;)11 193 5/'15(∫ ∞
−∞
|H(f)|2 ·N0df = N0
∫ ∞
−∞
h2(θ)dθ
72((
S
N
)
outt=0
=
[∫ ∞
0
u(−θ)h(θ)dθ]2
N0
∫ ∞
0
h2(θ)dθ
=
[∫ ∞
0
U∗(f)H(f)df
]2
N0
∫ ∞
0
|H(f)|2df
6-21-71 3-3- ( SN
)
outt=0
/-3) 7 H(f)
(,h(θ)
.:--; (,1 .-',:-2 .7'1 ,81 3-53.(-42;1(4 . (-84 :(;2 &'(() 2) (-((-)3 -, .'(79.
(
∫ T2
T1
f1(t)f∗2 (t)dt
)2
≤∫
|f1(t)|2dt ·∫
|f2(t)|2dt
&'(() 2) (-((-)3 -,1 254: 72 6-. ('-') 3-8'(; ('; *+41α) f2(t) = αf1(t)
'),7 (-((-)2 ;(3 (-((-)3 -, (2-2 * --(0 -531((
S
N
)
outt=0
≤
∫ ∞
0
(u(−θ))2dθ∫ ∞
0
h2(θ)dθ
N0
∫ ∞
0
h2(θ)dθ
=1
N0
∫ ∞
0
(
u(−θ))2
dθ
'),7 (-((-)7 *--4.- 3. 39 (:254) (-((-)3 -,) &'(() 2) (-((-)3 -,1 5(: 7 (17h(θ) = αu(−θ)6-7 :,3 '-83 -52
u(t)2) -,'3 .:(1. , (3 -21 -0;(,3
h(t)3:413 ,71
.1, (.1 .::1 ( . (, 6 '(-,6.1, (.13 .::13 ,'4: -21 -0;(,3
h(t))1.)3 193 5/'15 -(0 -55 )1.)32 * (415 6'+.3 5/'15 -(0 -5 ( 193 5/'15 -(0 -5 -; (1 *+(43 +(15 3'3.'1.3 - (αU∗(f)
,(3 -21-0;(,3H(f)
) &'(() 2) (-((-)3 -, .'95 2542 .:1 2 '+.3 5/'15 -(0556αu(−t) , (3 -21 -0;(,3
h(t)) 3,8(.3 ., .-:) 5 (17 254: 37 (;3 3-' (;
6. -',:-2 3'52,2 7 ( .('5.33 .'(.2 . (' ()43 . ('(79. ( . ('+3 (,: 39 4';5
6ω .(+(4 : 2) (, , (3Ω* +13 5/'1
8.1
6-( : 2) . (-');,3 . (,8(.3 27 (, 27 * +13 5/'1 2 5 ()/: .-5-0 -, (0 :-,2) -. ('-') (, '/5: .(-');, . (,8(. )) ) -) ((7 63 -5 (4 .4-'9 2) 2+(1 . (:52 * -8 (') /-::
8.2 6a, b, c, d, e, f (,3 ., '(/52 * 27 (: 61, 2, 3, 4, 5, 6 *-';13 (, 2)12 6* -18 3) -)* +1 5/'1 *-:(5 * -1;2 6'. (- 5(1 * +13 5/'1 3-3- 7 '. (- .75(1 3;(. 2) 2+(1 . (:52 * -8 (') 2776'. (- '/ (,1 52)5 2+(13 2) . (5/'3 ');,2 (, 0 (); 3:51 * 5/'1 -)32 -+7 .,9 &(/: '),1 '. (- 2(+ (,5 63+-/-3 21 4-3 2 .(+(4 :3 (, ., * +1 5/'17 '(/52 27 (: 302(' 2) 2+(1 . (:52 -+7
8.3 6213 2 047 302('5 3,8(. '(,1 27 ',.2 .-: 6302('5 . (-');, . (,8(. ) - '),1 . (+(4 : '. (- 39. (-84:(;3 27 (, . (-32 2(7 - * +13 5/'1
[0, 1]193 045 '511 3,-8- ) ' '(5 2+(1 . (:52 * -8 (' *,6
[0, 1]193 045 . (;-8'3
6* -,53 *-,:.3 ., . (1 --413 Ω2) . (8(54 .. 2) (, , (3
F.('(,13 (,
8.4
6'(,1 ,-3Ω2)
Ω38(543 .. (, '(,1 , (3 3'(4 +-1.) 3'413 '1(27
Ω ∈ F6
6A1 ∪A2 ∈ F
'1 (27 '(,1 , (3A1 ∪A2
* -9, .('(,1 *3A2
7 (A1
*, 66F2 --)
Ac (2) * -2)13 * -9,F2 --)
A*, 6
* -9,i = 1, 2, . . .
'(5Ai ∈ F
*, -01.1 (1 -5 6'(,1 ,(3 , .('(,1 2) . (:132 .-: +(/-, 66∪∞i=1Ai ∈ F
-:) σ-algebra
3'5 2, 31 - (, σ-field
3+) 31 - ,'4: 2-2 * -,:.3 27 ., *--413 . (8(54 .. (,6) (1 (. (,2 . (1)*, *-'9 * -,'4:
A,B.('(,1 (9 6∅ 5 1 (. '5+ 32-71 3:-,) 38(543 '1 (27 34-'3 38(543
8.5 6A ∩B = ∅ *, '1(27 . ()1 '5+ *32 -,
., .1--41( F
5 '(,1 27 '(5 .'+(13 3-84 :(; , -3P A .('5.3 (, .('5.3 .-84 :(;
8.6
*-,53 *-,:.36A ∈ F
2720 ≤ P A ≤ 1
66P Ω = 1
6*--4.1
i 6= j*,
Ai ∩Aj = ∅ '1 (27 *-'9 *3) Ai, i = 1, 2, . . . .('(,1 2) 3-:1 5 (, 272 6P
∞⋃
i=1
Ai
=
∞∑
i=1
P Ai
6P A+ P Ac = 1
*--4.1A
'(,1 272 -7 5(: ( 16. ('5.3 5/'1 *-,'(4 Ω,F,P 3)2)2 8.7
.('5.33 -31 2(,)2 ');, '5 131 3,-8-3 ) ' '(5 6* -0 (); . ('(,1 (:',.8.2–8.3
.(,1 (+58.8
2913 3:.)1 ., ) '3 -23. , (3n(t) = n(t, ω)
'),7 ∫ 1
0 n2(t) dt ≤ 3
2)12 -+1 32(+ ,2 3-3. 3-':,3)-+7 .-01.1 . ('-39 '. (- 3) ('+ ,7 6-;)1 ( *--4 , (3 , '. (- 0 (); 3-3 - (1 -3) -+7 227 '+5 0 -1) :ω6.'+(1 3:-, . ('5.33 -9, ,2 *, -7 '(,1 (39 7,) ,+((2
a-)11 ';1 27 '(5) 7
Ω* +13 5/'1 2
X(ω)3-84:(; , (3 , 1 '(8-45 -,'4, 3:.)1
8.9 6'(,1 ,-3 ω : X(ω) ≤ a 38(543*-1;2 31 (+ 3'(85 6
X(ω)2 * -:((7.1 '),7
X*()'2 25(41 '1(27
ω3:.)13 ., 0-1)32 25 (41) 52 * -)6ω : X(ω) ≤ a '(,13 ., ',.2 * -:((7.1 (:, '),7 X(ω) ≤ a 2)12 '(8-45 * ()':
) '3) /-::(n(t) = n(t, ω),
- (. : '511 3,-8-5 ) '3) /-:: 6X
5 -,'4,3 3:.)13 ., 1 :8.10
*-,53 , 13 ., '-+32 .-: -9, 6193 2) 3-84:(;7 -8'6X(ω) = n(2, ω) = n(t, ω)|t=2
,(3 V
02((5 ++1: -21)/ /.1 2)12t = 2
'5 '513 ) ' 2) '3 •6X(ω) =
∫ 1
0n2(t, ω) dt
,-3[0, 1]
195 * (/.5 ) '3 . (, 2) 3-':,3 2)12 '. (- * -57'(1 , 1 '-+32 ');, •6Y = max0≤t≤1 n(t, ω)
, (3[0, 1]
*(/.5 ) '3 . (, 2) -21-413 '3 •
,+((2 ) - -,'4, 3:.)1 ,(3X(ω)
) .(,'32 -+7 , 6Ω
5 38(54 ,-3 ω : X(ω) ≤ a -7 ' ('5 .-01.1 3'36. -/7 (:3 .'15 ,2 , .() 32 .-: 39 27 6F2) 3'+32 /--.32 ) - 7 *)2 6'(,1 (39 -7
1(
05 * -7' .25413 -)11 3:.)1 2) 3-84 :(; , -3
X(ω)-,'4, 3:.)1 2) (2-;3 .-84 :(;
8.11 3.'+3(FX(a) = P X(ω) ≤ a
FX(a)(2-;3 .--84 :(; 6
X(ω)-,'4,3 3:.)13 2) (2-;3 .--84:(;5 *-4 ( (:,) 3391
X1-3 (9 3'+356+525
a3:.)13 2) 3-84 :(; ,-3
3/7 (3 ,22 FX(a)
(2-;3 . --84 :(; . (:(7.8.12
-1 -1 3;-8'( .+'(- ,2 .-:(0 (:(1 ,-3 (2-;3 . -84 :(; •*-7' 4' 2541 -,'4, 3:.)1 '1 (27 6FX(−∞) = 0
(FX(+∞) = 1
.-01.1 4-(+1 -'1 2 ,2 (1 -5 •6* --; (-9,
a1 < a2*, •
P a1 < X ≤ a2 = FX(a2)− FX(a1)
63-5(4 2) (2-;3 .--84 :(; ., '--8 .(8-;45 32( ( *-04 -:; 2 3 (54FX(a)
-9, +-+5 3:.)1 , (3X
*,a272) 7
fX(θ)5 1 :) 1-' . -2-5'0:-, 3-84:(; .1--4 *,
8.13
FX(a) =
∫ a
−∞
fX(θ) dθ(8.1)
6X
, 13 2) -2(3 (2-;3 (, X
, 13 2) . (;-;83 .-84 :(; ,'4.fX
-9,
*--,'4, *-:.)1N
2) (, , (3N
+1-15X
-,'4, '(04 ((8.14
X(ω) = X1(ω), X2(ω), . . . , XN (ω)′
148
, (3 -,'4, '(04 (( 2) (2-;3 6Xi(ω)
'(04 ((3 -5-7' 272 291 '01'; (. (, *-5-7'3 272 .()1 ,(3ω
) 5 (17-,'4, 3:.)1 2) (2-;3 ) (1 2) 35/'3'(04 (( 2) 3-84:(; , -3
X(ω) = X1(ω), X2(ω), . . . , XN (ω)′ -,'4, '(04 (( 2) (2-;3 .-84 :(;8.15 3.'+3 (
1(
05 *-7' .25413
a = a1, a2, . . . , aN′ -)11FX(a) = P X1(ω) ≤ a1, X2(ω) ≤ a2, . . . , XN(ω) ≤ aN(8.2)
a272) 7
fX(a)3-84:(; .1--4 *,
8.133,' .(;-;8 .--84 :(; ) - -,'4, '(04 ((2
FX(a) =
∫ a1
−∞
· · ·∫ aN
−∞
fX(a) da1 · · · daN
- 3. (, 5)/2 .-:fX(a) =
∂NFX(a)
∂a1∂a2 · · · ∂aN
6. -0 -00 . (2. -,8.16 6
P A ∩B = P A · P B *, .5 .-0 -00 *--(2. -.25 * -,'4:A,B
. ('(,1 (9ω :
.('(,13a, b
*-';1 (9 272 *, .5 .-0 -00 *--(2. -.25 *-,'4:X,Y
*--,'4, *-:.)1 (9* --4.1a, b
272 *, .5 *3X,Y
*--,'4,3 *-:.)13 .('/, *-2-15 6 .5 *3 ω : Y ≤ b( X ≤ aP X ≤ a, Y ≤ b = P X ≤ a · P Y ≤ b
3:() (1 -5 (,FX,Y (a, b) = FX(a) · FY (b)
*--4.1a, b
272 *, .5 *-,'4:X,Y
*-'(04 ( -:) 339 3'(85FX,Y (a, b) = FX(a) · FY (b)
6 .5 *3 * X, g(Y )
-9, 2'(5 .--84 :(; *-4+4+12 ,-3)27 .-0 -:-1'0+ 3-84:(; , -3g( .5
X,Y*,
149
3'(85 .'+ (1 .2/ (.3 6EX = X = mX
. ('(831 ./,5 1 :X
-,'4, 3:.)1 2) .2/(.3 .,8.17 63,53
) -,:.5 .'+(1 .2/ (.3 6αi, i = 1, 2, . . . 2)12 *-+-+5 *-7' 25413 -,'4, 3:.)1X
-3 - •∞∑
i=1
|αi|P Xi = αi <∞
,-3 .2/ (.3 -9, *--4.1 39 -, :. *, (EX =
∞∑
i=1
αi P Xi = αi
) -,:.5 .'+ (1 .2/ (.3 -9, 6. -2( .(;-;8 ) -X
3:.)12) /-:: •∫ ∞
−∞
|α|fX(α) dα <∞
,-3 .2/ (.3 -9, *--4.1 39 -, :. *, (EX =
∫ ∞
−∞
αfX(α) dα
2) 3/ (:5 *-)1.)1 +-+5 3:.)1 -+- 2 3:.)13 ., *-5'41 6* -5 ('4 -+- 2 .2/(. * -'-+1 -227 ;(,5 •k(n)
2(−n, n)
043 2) .-;( 34 (2/ '/5:n';1 272 0 ('-; '. -5 65 ('-43 ., *-';)1 ( +-+5 3:.)1*--4.) 34 (2/31 ) ('+: 6* -04
−n = α(n)1 < α
(n)2 < · · · < α
(n)i < α
(n)i+1 < · · · < α
(n)k(n)−1 < α
(n)k(n) = n
maxiα
(n)i+1 − α
(n)i → 0
*--4) -,:.5 .'+ (1X
-,'4, 3:.)1 2) .2/(.3 6n1 '. (- '31 2+-
k(n)) -'8 7 *)2
n→∞ '),7) 7
B (54
k(n)∑
i=1
|α(n)i |
[
FX(α(n)i+1)− FX(α
(n)i )]
< B
,-3 .2/ (.3 -9, *--4.1 39 -, :. *, 64 -;1 2(+n272
E[X ] = limn→∞
k(n)∑
i=1
α(n)i
[
FX(α(n)i+1)− FX(α
(n)i )]
(8.3)
.(,53 . (' (831 ./,5 2'0:-,7 39 2(5 *-:11 * (73 .'(8 225E[X ] =
∫ ∞
−∞
αdFX(α) =
∫ ∞
−∞
αFX(dα)
6Stiltjes-Lebesgue
52 82-0 2'0:-, ,'4: 397 2'0:-,-,'4,3 3:.)13 '),7 . (1+(43 . ('+32 339 3,8(. .:. (: .2/ (. 2) .-2273 3'+33) 4 (+5
8.18 6. (;-;8 ) - -,'4,3 3:.)12 '),7( *-+-+5 *-7' 2541-9, 6
Y = g(X)-,'4, 3:.)1 '-+:
g3-84 :(; (
X-,'4, 3:.)1 .:35
8.19
E[Y ] = E[g(X)] =
∫ ∞
−∞
αdFY (α) =
∫ ∞
−∞
g(α) dFX(α)
.'95 5 () -/3 ., 852 ');, (2) (2-;3 ., 5)/2 '(8 -,Y
3:.)13 2) .2/(.3 ., 5)/2 -+7 '1 (276X
2) (2-;3 .-84:(;-2( (2-; . (-84:(; * , 1 '(5 31 (+2 6.2/ (. '-+32 .-: +-1. ,2( .-;( .2/(. ) - , 1 272 ,2
fX(α) =
0 α ≥ 0*,
2/π
1 + α2α < 0
*,fY (α) =
1/π
1 + α2 -7 *--4.1∫ ∞
−∞
αfX(α) dα =
∫ 0
−∞
αfY (α) dα = −∞
) ((-7 .,9 .1(2 6. -; ( -, ,-3 .2/ (.3 *2(, X
3:.)13 '(5 .2/(. '-+32 ');,) 7∫ ∞
0
αfY (α) dα =∞
6.2/(. '-+32 227 .-: ,2Y
'(5 72(
151
3-84:(;3 .,IA
5 1 : 6ω 6∈ A *, 4'( *,
ω ∈ Ac '1 (27 *-2)13 '(,13 .,Ac5 1 :
A'(,1 .:-35
7 '+(13 -,'4, 3:.)1 (39 6A
'(,13 2) .:--813IA(ω) =
1 ω ∈ A *,0
6ω ∈ Ac *,
6.2/ (.3 . (:(7.8.20
A
'(,1 272 6E[IA] = 0 · P Ac+ 1 · P A = P A
6EX = C
-9, -,'4, (:-,) (54X = C
*, 6-9, 6* - (54 (9
a, b(-3 -( .2/(. ) -
X,Y*-:.)12) /-:: . (-',:-2 6
E [aX + bY ] = a · E[X ] + b · E[Y ]
6(,2 *, -5 ( . -0 -00 *--(2. *-:.)13 *, -5-9, .2/(. *32 ) -( .-0 -00 *--(2. -.25
X,Y*-:.)13 *, 6
E[X · Y ] = E[X ] · E[Y ]
6P X = Y = 1
*, 4' 7.- (-(() '),7 E[X ] ≥ E[Y ]
-9, P X ≥ Y = 1
(, X ≥ Y *, 6
-9, *--5 (-/ ( * -12) *-7' 2541X
3:.)13 *,8.21
E[X ] =
∞∑
k=1
P X ≥ k
39 3'412 .2/ (.3 .'+3 -;2 3/7 (3E[X ] =
∞∑
k=0
k P X = k
=
∞∑
k=1
P X = k+
∞∑
k=1
(k − 1) P X = k
= P X ≥ 1+
∞∑
k=2
(k − 1) P X = k
= P X ≥ 1+∞∑
k=2
P X = k+∞∑
k=2
(k − 2) P X = k
= P X ≥ 1+ P X ≥ 2+
∞∑
k=3
(k − 2) P X = k
= P X ≥ 1+ P X ≥ 2+ . . .
=
∞∑
k=1
P X ≥ k
6* -0 :1 (1 * -,'(4 -,'4, 3:.)1 2) . (4 9/ 2) . (2/ (.2'-+: 6-,'4, 3:.)1
X-3 -
8.22
E[X2]-:) 0 :1 (1 •6
E[Xn]n'+1 0:1 (1 •6
E[|X |n]n'+1 02/ (1 0 :1 (1 •
E
[
(
X −X)n] n'+1 -97'1 0 :1 (1 •
σX =√
Var(X)4.3 .--0 6
Var(X) = E
[
(
X −X)2] :,-' (( .(:() •
6E[
X2]
= 1(
E[X ] = 0*, 21'(:1 ,'4:
X, 1 •
, 13 -9, -; ( -:) 0 :1 (1 25 ,(3) 27 , 1 , (3X
*, 3'+33 -;2Z =
X − E[X ]
σX
6
621'(:1 , 1 , (35('-4 ,-3 (),' 0 :1 (1 .2/(.3 -:) +81 6(2) * -0 :1 (13 27 ., 5)/2 ');, , 1 2) . ('5.33 4 (/ (.1* -3 (5 *-0 :1 (1 6-0 -:-1'0+3 5 ('-43 5-5 '(9-;3 (31 ',.1 -:)3 -97'13 0:1 (13 ( , 12 '-5 -0 -:-1'0+6 (2-;3 2 (: +-1 * -4;1 '. (-4 (/ ., *-'-+1 *-0 :1 (13 -9, -+1 '31 32( (:-,
(E[|X |n])1/n) -:703 -,:.3 *--4.1 *, -227 ;(,5 3'36-1)1 +/ ;(,5 (2-;3
6* --,'4, *-:.)1 (9X,Y
(-3 -- '+(1 , 1 (9 2) :, -' ((43
8.23
Cov(X,Y ) = E[(X −X)(Y − Y )]-.251 2-+532 .-',:-2 * --(2. -.25 (, *-1, (.1 -.25 (, 3-82'(4 -'/ *3 , 13) '1,:Cov(X,Y ) = 0
*,- '+(1 , 13 -5ρ*,.-13 *+41 (, 3-82'(43 *+41 6.-0 -00 *--(2.
ρ =Cov(X,Y )
√
Var(X)Var(Y )
/-:: 8(115 -(2. (:-, *,.13 *+41) ((-7 6|ρ| ≤ 1-7 3,': 353 6
ρ = 0-9, .-',:-2 * --(2. -.25
X,Y*,('(5
λ';13 2) '3 ., );/:(
E[(X−λY )2]-5 (-/3 -(0 -55 :(5. : 5 () -/3 '(82 6
08(11 *-:.)13 -:)2)'(5 ) (1 * (1-:-13 -7 254 : 3'-931 62 3 (() :(
λ-;2 ' (9: * (1-:-1 , (812 -+7 6-21 -:-1 3-3- -(0 -53
λ∗ =Cov(X,Y )
Var(Y ) 254: 39 ' 5-8: *,0 ≤ E[(X − λ∗Y )2]
= E[X2]− 2λ∗ Cov(X,Y ) + (λ∗)2 E[Y 2]
= E[X2]− (Cov(X,Y ))2
Var(Y )
) ,71 (Var(X)Var(Y ) ≥ (Cov(X,Y ))2 6) '+:7 |ρ| ≤ 1
(,*)2 --2 *-2(7 - * -0 :1 (1 +8-7 3,'1 39 ')4 6. (-('5.3 -52 *-0 :1 (1 -5 *'(;1 ')4 ',.: *--:5 * (7-76. (1--(1 . (-('5.3 5('-4
154
50 -3 '+ (1E[g(X)]
-7 (:5 ) ('+: 632( ( . -5 (-/ 3 -84 :(;g-3. ( , (3)27 -,'4, 3:.)1
X-3 -
8.24
α';1 272 -9, 6-; ( (
P X ≥ α ≤ E[g(X)]
g(α)(8.5)
2 5()/2 ');, 3-8-, (0 :-,3 .5(02 62'0:-,7 .2/ (.2 8.3
3,' -2273 (1 -5 (9 3/7 (35 )1.) : 3/7 (35)/:(α., 54: 6
fX(x) dx2 '(8-47
dFX(x)-(0 -53
E[g(X)] =
∫ ∞
−∞
g(x) dFX(x)
≥∫ ∞
−∞
Ix≥α(x)g(x) dFX (x)
≥∫ ∞
−∞
Ix≥α(x)g(α) dFX (x)
254:( .2/ (.3 . (:(7.5 )1.): .7 632 ( 3-84:(; , -3 -7 -:)3 ( .-5(-/g-7 *--4.1 (),'3 (-(()3 -, '),7
E[g(X)] ≥ g(α)
∫ ∞
−∞
Ix≥α(x) dFX (x)
= g(α) P X ≥ α
(::0) -;7 72(P X ≥ α ≤ E[g(X)]
g(α)
6Markov
5(4'1 0;)1 ., 32-/. ,-5: 6. (1'(;1 . (:41 ';1 ) -432 ');, 39 0;)11*--4.1
α > 0272(
X-5 (-/ -,'4, 3:.)1 272
8.25
P X ≥ α ≤ E[X ]
α
'-+32 ');, -5 (-/X
) ((-7) 52 (1 -) *-4+4+12 6-5 (-/X
) ((-7 g(x) = x
3'-/531 +--1 .5 (: (9 3 :0666.2 (. .'/ ,-3 3:03 .-;( -, .2/ (.3 *-,) 5(17 6-, :. 27 ,22 .2/ (. (26Chebyshev
5)-5 8 */ .,'4: 39 0;)11 .5(:3 .; (: 3 :0
*--4.1 -5(-/α272(
X-,'4, 3:.)1 272
8.26
P |X | ≥ α ≤ E[X2]
α2
P |X − E[X ]| ≥ α ≤ Var[X ]
α2
-5 (-/α272) ((-7 0;)131 .-+--1 .5(: 3 :(),'3 3'()3
P |X | ≥ α = P
|X |2 ≥ α2
6X − E[X ]
-,'4,3 3:.)13 2 0;)13 .2;3 - :, -' (( 2) 3'+331 .5(: 3--:)3 3:03(6Chernoff
5(:' 8 */ , (3 0;)1 (. (,1 5(:3 .-/- )+/ */*--4.1
θ ≥ 0272(
α272
X-,'4, 3:.)1 272
8.27
P X ≥ α ≤ E
[
eθ(X−α)]
6.1 --4 -1 - +85 .2/ (.3) -,:.52542( 0;)13 ., 2-;32 ');, 72 632 ( ( . -5 (-/ 3-84 :(; , -3
x3:.)13 2)
eθ·x 3-84:(;3) 52 * -) :P X ≥ α ≤ E
[
e(θ·X)]
eθ·α
= E
[
eθ(X−α)]
- .'+(1 , -3 6ν-)113 3:.)13 2) 3-84 :(; , -3
X, 1 2)
φX
.-:--;,3 3-84:(;3φX(ν) = E
[
eiνX]
=
∫ ∞
−∞
eiνα dFX(x).
fX(α). (;-;8 -2 ( (2-; ) - 3:.)12 *,
φX(ν) =
∫ ∞
−∞
eiναfX(α)dα
.-:--;,3 3-84:(;3 3'41 275 -7 . (,'32 ');, 6fX(α)
2) 3-'(; .'1.3 2) 57 ('13 +(183 , (3φX(ν)
'1 (276FX(·) (2-;3 . --84 :(; ., .-1)1 +/ 3'-+1
φX
-7 .1--4 +-1. 72( 50 -3 .'+ (1 .-:--;,3 3-84:(;3 -7 52 *)∣
∣eiα∣
∣ = 1
.(4 9/ '(0 -+- 2 .-:--;,3 3-84 :(;3 ., 8--2 .-: -9, '+ 271 * -01 (1 * -1 --4 *, 6α272
φX(ν) =∞∑
k=0
(iν)k
k!E[
Xk]
*--:70 * -,:. ./. '),7φX(0) = 1
∂φX(ν)
∂ν
∣
∣
∣
∣
ν=0
= iE[X ]
∂kφX(ν)
∂νk
∣
∣
∣
∣
ν=0
= (i)kE[
Xk]
6* -0 :1 (13 2) 5() -/ .');,1 .-:--;,3 3-84 :(;3 . -+- 0';5) 7
-+- 2 .'+(1B
'(,1 .:35A
'(,1 2) . ('5.338.28
P A | B =P A ∩B
P B
2) . (:(7.3 27 ., 32 ) - 72( B
38(545 .97 ('13 A
2) 3-84 :(;7 .('5.3 -3 (9 (54B
'(5 -7 52 * -)3'+331 . ('-) - .5 (:( +,1 .-) (1 -) 3,53 3:03 6
8.63'+3 .('5.3
*--4.1P Ak 6= 0
*-1--413 Ak, k = 1, 2, . . . ,K .('(,1 '(58.29
P A1 ∩A2 = P A1 | A2 · P A2
P A1 ∩A2 | A3 = P A1 | A2 ∩A3 · P A2 | A3
P
∩Kk=1Ak
= P
A1 | ∩Km=2Am
· P
A2 | ∩Km=3Am
× · · ·
× P AK−2 | AK−1 ∩AK · P AK−1 | AK · P AK
P
∩K−1k=1 Ak | AK
= P
A1 | ∩Km=2Am
· P
A2 | ∩Km=3Am
× · · ·
× P AK−2 | AK−1 ∩AK · P AK−1 | AK
* ()'2 .-: -9, 6B = ω : X(ω) = β '(,13 ., (54
β'(5 '-+:( *--,'4, *-:.)1 *3
X,Y-7 .7 /-::
P A | B = P A | X(ω) = β =P A ∩B
P B+8-7 6P B = 0
;, ,-3B
'(,13 2) . ('5. -33 .(;-;8 ) -X
3:.)12 *, 2)12 *-5' * -'4 -15 *2(,.-:. (13 . ('5.33 ., 9, '-+:
'-+:X
-,'4, 3:.)1(A
'(,1 '(58.30
P A | X(ω) = β = limε→0
P A | β ≤ X(ω) ≤ β + ε
6. (1--4.1 . (:(7.3 27) 7 .-:. (1 . ('5.3 '-+1( *--4 +-1. 39 2(5 ) /-:: (:,7 '+ (1
X3:.)13 .:35
Y3:.)1 2) 3:. (13 (2-;3
8.31
FY |X(α | β) = P Y ≤ α | X = β
62 (53 '+ (. (, '-+: .(;-;8 * 3:.)1X
*, 63212 '+(3 -1 - +86+-1 27 3;- (1 3:-, 3-:.33 -9, .-0 -00 *--(2. -.25
X,Y*,
-9, * --(2. -.25 *3A,B
.('(,13 *-,8.32
P A | B = P A 6 -9, .-0 -00 *--(2. -.25 *3
X,Y*-:.)13 *, 72
FY |X(α | β) = FY (α)
-9, .5A,B
*, .-:. (1 . ('5.3 .'+31 . (5 (: . (:03 -.)P A | B =
P A ∩BP B =
P AP BP B = P A
63 :. (13 (2-;3 .'+31( 71 .5 (: 3-:)3 3:03) 7
f3-84 :(; .1--4 *,
8.33
FY |X(α | β) =
∫ α
−∞
fY |X(θ | β) dθ
6X
.:35Y
2) .-:. (1 . (;-;8 (, 3:. (1 -2( (2-; ,'4.f-9,
.(;-;83 ., 5)/2 ');, -9,fX,Y (α, β)
.;. ()1 . (;-;8Y
(X
2 ) - *, -7 5 (: . -:. (1 . (;-;8 2) 3'+3312327 .-:. (13
fY |X(α | β) = limδ→0
FY |X(α+ δ | β)− FY |X(α | β)
δ
= limδ→0
P α ≤ Y ≤ α+ δ | X = βδ
= limδ→0
[
limε→0
P α ≤ Y ≤ α+ δ, β ≤ X ≤ β + εδ · P β ≤ X ≤ β + ε
]
=δ · ε · fX,Y (α, β)
δ · ε · fX(β)
=fX,Y (α, β)
fX(β) -9, .;. ()1 . (;-;8 ) - *, *(7-2fY |X(α | β) =
fX,Y (α, β)
fX(β)(8.7)
fY |X(α | β).-:. (13 . (;-;83 (
FY |X(α | β)3:. (13 (2-;3 * 7 (
P Y ≤ α | X = β .-:. (13 . ('5. -335(17 254.- 6
X-,'4,3 3:.)13 .,
β';13 * (415 5-832 .7 .-: 6
β3:.)13 2) . (-84:(; *2(7 *3-+- 2 '+ (13 ω
5 -(2. )+/ -,'4, 3:.)1P Y ≤ α | X = P Y ≤ α | X = β|β=X
(-(()3) 5(: (9 3'+31FY |X(α | X) = P Y ≤ α | X = β
6fY |X(α | X)
2) . (1)13 ., -532 ) - 31(+ 3'(85 6X(ω) = β
*--41ω*, 4'( *, *--4.-
'+5 .-:. (1 .2/ (. '-+32 ');, 2-' (2-; 2) . (:(7.3 27 25 , (3 3:. (13 (2-;3 (54β'(5 '(1,76.2/ (. (:'+3)
-+- 2 .'+(1β
'3 ., 2541X
3:.)13) .:35Y
-,'4, 3:.)1 2) .-:. (13 .2/ (.38.34
E [Y | X = β] =
∫ ∞
−∞
αdFY |X(α | β) =
∫ ∞
−∞
αFY |X(dα | β)
-9, 3:. (1 -2( (2-; ) - * -,E [Y | X = β] =
∫ ∞
−∞
αfY |X(α | β) dα
*--4.1g,-3)27 3-84 :(; ' (5
E [g(Y ) | X = β] =
∫ ∞
−∞
g(α)FY |X(dα | β) =
∫ ∞
−∞
g(α)fY |X(α | β) dα
63 :. (1 -2( (2-; *--4 ( 3+-15 * --4.1 ('/,3 (-(()3 '),73'415 (17 72(
β3--:.33 3:.)1 2) 3-84:(; , -3 (:'+3) .-:. (13 .2/ (.3 32-' .2/ (. 2) 3'415 (176
β*(415
X3:.13 3:.)13 ., 5-832 ');, *-:. (1 .(;-;8 (, (2-; . ('5.3 2)
E [Y | X ] = E [Y | X = β]|β=X.-:. (13 .2/ (.3 6'. (- .4 -(+1 ( '. (- .0);(1 ,-3 .-:. (1 .2/ (. 2) .4-(+13 3'+33 *-4+4+12 3'33-3- . -:. (1 .2/ (. (5 3 -; (1) (-(() 27 72 ;, (. ('5.3) '(,1 -+7 + 4' .'+ (1 -,'4, 3:.)-17 6(9 3+(4 :2 /--. : ,2 )135 6;, 3. ('5.3)ω.8(542 0'; (7 :
.2/ (.2 ) - *2(, .2/(.3 . (:(7. ., .)'(- , -3 72 632-' .2/ (. (17 .5) (/1 .-:. (13 .2/ (.3 (54β'(56. (-+(/- . (:(7. * .-:. (13
'(,.5 6. -; ( .2/(. (2 ) -) 3:.)1 272 .'+ (1 .-:. (13 .2/ (.3 6. -:. (13 .2/ (.3 . (:(7.8.35 .2/(.2 31(+5 6.1--4 3-3. .-:. (13 .2/(.3) -+77 *-) ('+3 * -,:.3 *-1--4.1) +-1. /-:: 2-2 . (:(7.3
32-'3X
-,'4, 3:.)1 272(A
'(,1 272 6E[IA | X ] = P A | X
6E[Y | X ] = C
-9, -,'4, (:-,) (54Y = C
*, 6-9, 6* - (54 (9
a, b(-3 - . (-',:-2 6
E [aZ + bY | X = β] = a · E[Z | X = β] + b · E[Y | X = β]
E [aZ + bY | X ] = a · E[Z | X ] + b · E[Y | X ]
6E[Z | X = β] ≥ E[Y | X = β]
-9, P Z ≥ Y | X = β = 1
(, Z ≥ Y *, 66. -:. (13 .2/(.2 . (+/ (-1 3 . (,53 . (:(7.3
6E[X | X ] = X
* 72(E[X | X = β] = β
63)27
g, h.(-84:(; ' (5 6
E [g(X) · h(Y ) | X = β] = g(β) E [h(Y ) | X = β]
X,Y*-:.)13 -:)5 3--(2.
h*, '. (- -227 ;(,5(
E [g(X) · h(X,Y ) | X = β] = g(β) E [h(β, Y ) | X = β]
31,.35 *-'413 -:)5 * --4.1 .('+33 -;2 72E [g(X) · h(Y ) | X ] = g(X) E [h(Y ) | X ]
E [g(X) · h(X,Y ) | X ] = g(X) E [h(X,Y ) | X ]
6E [E(Y | X)] = E[Y ]
-9, .-0 -00 *--(2. -.25X,Y
*-:.)13 *, (E[Y | X ] = E[Y ]
6E [g(X) · h(X,Y )] = E [g(X) · E(h(X,Y ) | X)]
* (547X
2 /--.32 ');, *:1, 9, *--(1 '2 54: 3:.17X
2) (7' *-, -7 .'1 (, 3:(7. .('36342/3 .(,'4: ( .(:(7. 63:.17 4' ,2 3:. (17 (+-4;.5(2-;3 .'95 3,': 3:(7. ., 63.2+ , (3 (18 *
X2) .()13 (2-;3 7) .('+331 .5 (: 3:(7.* -:() * -+-4;. -5 +-';32 ');,. (9 3'+3 6
Z = X,-3 (.'+3)
Z)+/ 3:.)1 '-+: 3/7 (33 '(82 63:. (13-7 . (,'32 (:-2 )+/3 (1 -5 6
X3:.)13 (. (, 2)
E [g(Z) · h(Z, Y ) | X = β] = g(β) E [h(β, Y ) | X = β]
3'+33 -;2E [g(Z) · h(Z, Y ) | X = β] =
∫ ∞
−∞
∫ ∞
−∞
g(z)h(z, y)FY,Z|X(dz dy | β)
=
∫ ∞
−∞
g(β)h(β, y)FY,Z|X(dy | β)
7X = Z
5 97'.1 (2-;3 .-:. (13 . (;-;82 3/ (:3 ., 0';5 3,' .()13 3:. (13 (2-;3 .'+31 ((-71, (3 .-:. (13 (2-;3 . --84 :(; . :-/51 -0 :((2' (:-,Z3:.)13 39 52)5 63.2+ .-84:(; , -3 .-:. (13 . (;-;83)
*, 62'0:-,2 &(/1 2, (, -8 (32 ');, 72( 2'0:-,3 .:-/51 (54 ,(3g(β)
(:5 6* -0 :1 (',3 2 -;)1 (:-,7
E [g(Z) · h(Z, Y ) | X = β] = g(β) ·∫ ∞
−∞
h(β, y)FY |X(dy | β)
= g(β) E [h(β, Y ) | X = β]
6. -:. (13 .2/ (.3 .'+31 5 (: ('/,3 (-(()3 '),7( 8.7)
3/ (: ., 0';5 3,' 3:. (13 (2-;3 . (:(7.1 5 (: 5 (),'3 (-(()3 63'+331 *-5(: 5 * -:(-(()3 ',)6. (1+(43 . (:031 5 (: (52 68.32
3:01 5(: -:)3 (-(()3 6. -:. (13 . (;-;83 '(5
