С Л Семаков - Элементы Теории Вероятностей и Случайных Процессов

. .

-

2011

519.21

22.171

30

. . -. - .: , 2011. - 232 . - ISBN 978-5-9221-1345-8.

. , .

- - - . .

ISBN 978-5-9221-1345-8

, 2011

. . , 2011

1. ........................................ 6

1. ............................................................... 6

.............................................................................................................. 9

2. ..................... 14

............................................................................ 17

3. , . . 30

. . . . ................................................................................................... 35

4. ................................... 41

...................... 43

5. .............................. 50

.............................................................................................................. 54

6. . . . ..................... 64

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7. ......................................... 90

.................................. 93

8. 109

. ............................................................................... 113

2. .................... 123

9. , . 123

................... 128

10. ,

................................................................. 134

............................................................................................................... 136

11. ................................ 142

................. 143

12. -

............................................. 155

............................................................................................................... 161

13.

.................................................................................. 168

................................................................................................... 174

14. ........................ 189

.......................................................................................... 200

15. ............................................... 207

............................................................................................................... 214

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r*k _

k \ (n - k )V

k

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k ,

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k - (ti (k 1)) .

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k\Ckn = ( - 1)( - 2 ) . . . ( - ( - 1)),

_ ( ~ !)( - 2) . . . ( - (; - 1))

L n ~ k i ~ ~

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p k _ ~ k \ (n - k ) V

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2. . ,

1, 2, . . . , ?

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k, k < , -

A* = n ( n - l ) . . . ( n - ( k - 1)) =

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A kn = CknPk. 3. 5 . -

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j ! -

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1. -

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2k . -

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2 + 22 + 23 + 24 + 25 = 2 + 4 + 8 + 16 + 32 = 62.

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1, 2- r

( -

) 9 10 10 10 = 9000,

1 9,

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9 , ,

10 .

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, 3:

8 9 9 9 = 5832. , -

, 3, N = 9000 5832 = 3168.

. N\ -

, 3,

N2 3, / 3, N 4

3. N = N\ + N 2 4- /3 + N 4 , , , N 4 = 1.

N\.

3-----, ()

3. , ,

9 9 9 = 729. , Ni

-3 , 3 - , -----3, -

, 3,

, -

0. -3

8 9 9 = 648 , 3 - -

3. , N\ = 729 + 3 648 = 2673.

N 2 . -

33 , 3 -3 - , 3 3, -3 3 -, -3 -3 , 33,

-

3, , -

0. -

33 , 3 -3 - 3 3 9 9 = 81 ,

-3 3 - , -3 -3 33 8 9 = 72 . ,

N 2 = 3 81 + 3 72 = 459.

/3. 333-, 33-3,

3-33 -333. 9 ,

-333 8 . , /3 = 3 9 + 8 = 35.

,

N = + N 2. + / + N 4 = 2673 + 459 + 35 + 1 = 3168.

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10. Pj 'p^2 Pi,p2 , ,Pk ?

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< , m2 < 2 , .. , rrik < . -

(\, 2 , -

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mi 0,1,2,

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\ m2 ( + 1) -

, . . . , mi,m 2 , ...,/re*_i

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( + 1)(2 + 1) . . . (n* + 1).

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Aix,Ai2, . . . , A i m.

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2 , $ 6.

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h = 4, = 6) {} = / = 3/6 = 1/2.

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{ + 2 + . . . + + . . . } = P{Ai} + P{A2} + . . . + { } + . . . .

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k, -

k < 10, 7, 8, 9, 10. , k = 7.

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N = 36550.

= 365 364 (365 - 49) = ^ 5.

JV tn ^ 365

N ~ ~ 365

365 364

365 ' 365

{ } \

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k .

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(k + 1 ) .

Nk+i.

1 2 3 k k + l k+ 2 k + 3 ' '

(-1)

X Y , ,

N k+1 = 2 ( - 2)! ( - (k + 3) +1) = 2 ( - 2)! ( - k - 2).'------ V------ '

2 ( (k + 3) + 1)

,

(k + 1) ; ( 2)! -

( 2) ( 2) .

2 N^+\ ,

X Y ,

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Nk+2 = 2 ( - 2)! ( - (k + 4) + 1) = 2 ( - 2)! ( - k - 3),

= 2 ( - 2)! ( - ( - 3 + 2) + 1) = 2 { - 2)! 2,

JV_2 = 2 ( - 2)! ( - ( - 2 + 2) + 1) = 2 ( - 2)! 1.

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k ,

= /&+1 + Nk+2 + + Afn_2 = 2 (n 2 ) \ - i =i=i

= 2 (n - 2)! 2 = 2 (n - 2)! l ? ~ k ~ 2 , ( n - k - 2 ) =

i = 1

= (n 2)! (n k 1 )(n k 2).

( - 2)! ( n - k - 1 )( - k - 2 ) _ ( n ~ k - 1 ){ - k - 2)

7. , 52 , 6 . ,

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: I, , III, IV.

10 :

1 I II, III, IV, I, I;

2 1 II, III, IV, I, II;

3 1 II, III, IV, I, III;

4 1 II, III, IV, I, IV;

5 I II, III, IV, II, II;

6 I II, III, IV, II, III;

7 1 II, III, IV, II, IV;

8 1 II, III, IV, III, III;

9 1 II, III, IV, III, IV;

10 - I, II, III, IV, IV, IV. Ni - , i =

= 1,2, . . . , 10.

N l = N 5 = N s = = ? \3 \ }3,

= N 3 = V4 = N 6 = 7 = N9 = Cf3 Cf3 C\ 3 C\3,

Ni + N2 + . .. + N10 4?|{} + ^^'' 133 38____

|2 " 562 - 4 7 .4 9 . 5 - 1 7 -

= 0,426 . . . .

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1/2 / / :

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2

10. N - . -

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. {\ + . . . + } - ..

{\ + . . . + } = ^ 2 P{Ai\} ~ P{AhAi2} +/ i = l \< i \< i2< n

+ P{AixAi2Ai3} - E P{AiiAi2Ai3Ai4} + . . .

1 < i 1 < /2 < i 3 < n 1 < i 1 < i2 < is < /4 < n

. . . + ( - l )"+1 Y , P{AkA h . . . A in}.\ < i X< i 2< . . . < i n < n

1 < i\ < /2 < h < - - - < n , -

; -

,

P{AhAi2 . . . A i m}, < ,

1

= 2,3,4. 2:

{ + 2} = {\} + {2} - { 2}.

= 3:

{ ! + 2 + 3} = { !} + {2} + {3} -

- {{ 2} + { 3] + {23}) + {\23}.

= 4:

{! + 2 + 3 + 4} = {!} + {2} + {3} + {4} -

- ({ 2} + { 3} + { } + {23} + {2 } +

+ {3 }) + { 23} + { 2 } + { 3 } +

+ {234} - { 1234}.

-

. = 2 .

= k. ,

= k + 1. = + 2 + . . . + *,;

, = A tA k+1, = 1,2, . . . , k.

{1 + 2 + . . . + Ak + Ah+1} =

= { + *+1} = {} + {*+1} - {*+1.} =

{ -

: , , D ( + C)D =

= BD + CD}

= { + 2 + . . . + Ak} + P{Ak+1}

P {A xA k+x + 2^ +1 + ... + AkAk+1} =/

= (EpW.}- E p (a M + E 1 + . . .i\=1 \

(S^W - ' />{.-,>+'=1 \< i\< i2 < k 1

, -

= k + 1. ,

.

. - , -

, i- , / = 1, 2, . . . , .

pr / | . i _ ( N - 1 ) ! _ 1 ^ ^ 1 _ (N - 2)! _ 1

1 .) - tN ~ 3)! - 1 , , 1 - ~ ;V(,V - !)f;V 2) ..................

1 - ) = ^ ~ ^ = __________ -__________1 12 " (m-r N { N

(1} , P{j4('jy42

= + + + = 1 >(=1 '-----------v---------- '

C'n = N

J 2 P{AiAj} = _ + . . . + 1 ) = 2!1 ^ v ^

>^9 iV! C t= -----------N~ (N2)! 2!

\ ' I I ! - 1\ < i l < i 2 < . . . < i m < N 4 1 111 v

C Z = iN (Nm)\m\

Y p {AhAi2 -AiN_{} + + + jjj (jv_ n r1

, ,

,

{\ + 2 + . . . + Av} = 1 ^ +

, 1 {\ + 2 + . . . + - }

(N-+ 1) * -

= 1:

N

P {A x + A 2 + . . . + A N} & 1 - -1 = 0,63212 . . . .

,

N 2/3.

. 30 8 , -

.

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2 , 2 1

.

. N 30

8 -

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30 , -

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. N .

8- ,

8 , -

8 . . ,

N = 8 - 8 - . . . - 8 = 830.'------V------ '

30

N . -

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4 ,

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12 ; 2

12

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\ /3 /^ *2/^ *6 /^3 8 5301>6 \U3Lr24L42,

= % = 1(8 4,0269 . . . .N

3. ,

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N

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A, N b , N^B .

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N b

N a b / N _ { }

N g / N { } '

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{ | } = {}, . . .

{} = { | } {} { \ } { } ,

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{} = { } { } .

-

\,2, -

, , , -

\ + #2 + . . . + (

, # # 2, . . . , # ).

= \ + AH

Hi, i = 1, 2, . . . , , -

, ,

.

1. , , 1-, 2- 3- 25, 35 40 % -

. 5, 4 2%.

.

, 1- ?

2- ? 3- ?

. :

i- , i = 1,2,3;

.

{} = 0,25; />{2} = 0,35; {3} = 0,4;

{ \Hi} = 0,05; { | 2} = 0,04; { | 3} = 0,02.

2,

, :

J2P{A\H,}P{H,)i= 1

0,05 0,25 25

0,05 0,25 + 0,04 0,35 + 0,02 0,4 ~ 69

, | ) =

,{\,}{,}

p |A} = ^ M L = | .

52{\,}{,}1=1

2. , , .

., b . -

,

q; + q = 1 ( ).

( , ) 1 .

,

.

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', + = 0 ( + ,

0 , ,

), Pq 1 ( 0,

).

- -

; .

{} = , { } = , {} = q, P { A \H } = Pn+h

{ \ } = _

( + q ) = +\ + Pn-\q, q{Pn - -1) = (+1 - ),

Pn+l - P n = ^ P n - P n-l).

{} = { | }{} + { | }{).

+ \ I" \'

1,2, . . . , + b 1 :

,

k = 0 k = 0

(/>+.- = E ( f ) V i - - (2)

, q , :

+ 1 - 0 = (Pi - 1 ) ^ -

+1

(-1 +i - 1 = ( - 1)- /-------

* - 1

= + b 1, :

-\-( i y _

1 = (\ 1) ---------, - 1

1 l - q-

I-' ' ( - 1'

1 _ J . m " ( l \ a+b_ ( S }

= ' ' 11 = / 11 = \ / >

" ( - ' I - ( -1 " ( -

+ / \ / \ ^

, ) -( - ~ - /

= q = 1/2 , -

(2),

Pn+i - 1 = ( + 1)( - 1).

= + 6 1

1 = ( + b)(P\ 1), , - 1 = - - , + 0

= ( - 1) + 1 = 1 - +

,

. = 1 - 6 + + '

,

b , ( = q = 1/ 2), -

,: = ~ 0 .

a + b

(3). Qt, -

,

, , , . .

> q. , , ( ~ ), -

(3) (4):

. .

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.

1. , -

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. . ,- , ,

- .

{\ 2 3 + 1 2 3 4 + ... + \ 2 . AfcA/i-t-i + ... |^ 4i}

_ P{A{A2Az + 1 2 34 + ... + \ AkAk+i + } _

{\23} + {{24\ + ... + {\ .. AkAk+\} + ...

(4)

{)

{)

2. , ,

> 1, = 1 - , -k=\

. ,

k > 1 .

. -

: , ( 1) , . . . , ( 1) -

, , . . ( + 1) .

k = 1,2, . . . ,

{ k | } =fi 1

,

{ k } =

= { (k + ) , k } =

= ^ 2 (^ k | ( + ) }

{ (k - ) }) =

. -4-\

= V 1 aDk+m = - = - V [+ dx =2 s k + m + 1 ^ k + m + l - J= 0 = 0 = 0

A Jk

- 0 0 0

( ~{ + ~ 2 + . . . + X + 1 + ^ dx =

( xk ~ 1 2 . | ,= ~ J \ T + r r -l + . . . + j + x + \ n \ x - l \ j

=

= 0

= Q/ i i '\ 1 - ~

3. ,

k , \ ke~x/k\ > 0.

-

. ,

5 .

. :

,

0 , 1 ,2 , . . . ;

,

0,1 ,2 , . . . .

P{HsBo + HS+\B\ + . . . + Hs+mBm + . . . } ^ 2 P{Hs+mBm} m=0

oo oo \s+ m A

^ 2 P{Bm | Hs+m}P{Hs+m} = ^ p)mpsm= 0 m= 0

o-A(s + m)\n \ s+me~x __ ps\ se~x ^ (A(l - p)Y

=0

E KS^my., _ )m s 'v ' e = P * em\s\ K P> P (s + m)\ s! ^ m\

m=0 m=0

x 2 3 ~ m

^ ==1 + v + 2_ + _ + = ^ P _+ + 2! + 3 ! + " ^ ml

m=0

,

v ((1~ )) = ^-)2 ^ ml= 0

ps\ se - x _) _ ps\ se~xP

s! s! '

. { | Hs+m}

(. , 4).

4. - . , 4- ,

; 2- 3-

, 1- -

; 1-

1- . Si = 33,7 % 1-, S2 = 37,5 %

2-, S3 = 20,9% 3- S4 = 7,9% 4- .

, -

. . ,, i = 1,2 ,3 ,4 , ,

- ; -

-

.

{) = ^ { \ 1}{,}.

1=1

P{Hi} = Si, i = 1,2,3,4, s, = Si/. , { \ } = si.

, 1- , -

,

1- , sj. ,

P { A \ H 2} = s2 + s x, P { A \ H z} = s3 + s h { \ 4} = 1.

,

{} = sisi + (s2 + si)s2 + (s3 + si)s3 + s4 =

= sf + s| + S3 + Sl(52 + S3) + S4 = sf + s| + Sg + 5i (1 Sj S4) +54 =

= s | + S3 + Sj + S4O Si)

- 0,140625 + 0,043681 + 0,337 + 0,052377 = 0,573683.

5. 4 , -

,

. . \ , ,

, 2 . ,

{\} = {2} = 0,573683 (. 4). ,

\ 2 , , -

. ,

{ + 2} = { } + {2} - { 2 } =

{ } + {2} - { } { 2} =

= 1,147366 - 0,3291121 . . . = 0,8182539

6 . 70% 30% - . 10% , -

5 % . -

. ,

.

. N .

0,7N 0,3,/V .

, 0,1 0,7N,

0,05 0,3/V. ,

0,1 0,7N + 0,05 0,3,/V , 0,1 0 ,7N -

. ,

0,1 0,7N _ 0,07 _ 14

0,1 0 J N + 0,05 0,3N ~ 0,07 + 0,015 - 17'

7. () , .

.

, -

/? .

qo ,

, q\ ,

.

. ,

, , .

qo = { \ },

_ _ { } _ { \ } { } _ (1 - )

~ {} ~ { \ } { } + { \ } { } ~ ( 1 - ) + ( 1 - / ? ) ( 1 - ) '

, q\ = { \ },

_ { } _ ( \ } { } _

qi ~ ( } ~ ( | } { } + ( | } { } ~ + ( \ - ) '

8. - 1,5 .

() -

2 , .

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; \

, #2 .

[ 1 { \ 1}

1 11 1 { \ 1} { 1} + { \ 2}{2}'

, N

.

/>{//,} = ^ , {2} = !L = .

= 1,5( ), {{\ = 1,5(1 {\}),

{\} = 0,6, {2} = 0,4. ,

{ \ 1} __

{ | 2}

,

^ ' > = = 0-75-

9. 1 /?.

.

,

7 . ,

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#2 , , .

1 _ { \ 1} { 1} _ -(1-7)

1 11 } P{A\Hi}P{Hl} + P{A\H2}P{H2} (1 7) + (1 /3) 7

10. 1- 10 , 8 ; 2- - 20 , 4 .

,

. , .

. ,

{} . Hi 1- 2-

, 2 1- 2- , ,

.

- / - 1 2 - * 4 10 20 100 { 2* 10 20 100

>{} = 1 - { 1} - { 2} = 1 - = ^ .

{} = { | Hi}P{Hx} + { | 2}{2} + { | 3}{3} =

16 16 1 68

4 .

. -

, .

{ } = , P{A} = l - p = q.

Pn(k) , -

k , k -

. ,

. , -

k ,

( k ) , pkqn~k. ,

Pn(k)

pkqn~k

k ( k) .

Ckn,

Pn(k) = Cknpkqn- k.

, -

.

() = - = ~ 2/2. ../27

,

" ~ -

k - n p = = . .

1 ( -). > , k -> \k \ 0 ,

, , \ / 2 > , \k \

Pn(k)lim = 1,

ti ()/ y/npq

. . > 0

1 - < w(% = < 1 + -4>(x)/^npq

-

.

Pn{k\ < & < &2) , -

- [6&2].

k=k2

Pn(k 1 < k < k2) = Pn(k).k=kl

1, -

. , -

( 1 -

-) .

2 ( -). -

Z\ Z2

^lim ( + zx^/npq < k < n p + Z2 yjnpq) = (2) - (21),

X X

() = [ (t) dt = - = [ e~ f2/ 2 dt.J v 27T j

00 00

, , -

.

3 ( ). > 0 > ,

p n(k) - ^ ~ - ,

= .

-

/>

1. 10 .

1/ 2, , :

) 5 5 ; ) 3

8.

. ,

= 10, = 1/2. (

, )

,

{ 5 5 } = (5) = =

- J 2!_ J _ - 63 5!5! ' 210 ~ 256

k= 8

{ 3 8} = ^ P\o(k) =

k= 3

= v r * n V ' V 0-* . ^^ 10 \ 2/ \ 2 / 1024k= 3

2. -

.

. k

Pn(k) = Cknpkqn- k

= 1 q, Pn(k) -

.

' pk+iqn-k-iPn(k + 1) _ (k + !)!( fe 1)! ____

Pn(k) n! k k

k \ (n - k ) \p q

_ k\(n k)\ p _ n - k p

~ (k + \ ) \ ( n - k - \ ) \ ' q ~ k + l ' q'

,

Pn(k + 1) > Pn(k) ( n k)p > (k + l ) q k < np q;

Pn(k + 1) = Pn(k) -- k = np - q\

Pn(k -I-1) < Pn{k) & k > n p - q.

, Pn(k) -

k ,

. q , -

Pn(k) -

k\ q q + 1 = + . - q -

, Pn{k)

k, ,

q. ,

&.. k, Pn(k) ; q , (1) -

..>

Pn(k), . .

k : kH_B, = q kH,B. = + .

3. 730 . ,

-

, 1/365 365 .

, 1 .

. ,

1 ,

.

&.. = 730 -

= 1/365 . ,

2:

.. = 2.

4. 3 , , 1 .

.

- q < &.. < + , (1)

- q < kH.B. < + ,

7(3 < k < 730) = 1 - 7(0 < k < 2 ) =

= 1 - -7(0) ~ ') - -7 (2).

,

'() = ~2, 7 30 ) = 2-2 , 73(2) = 2~2,

1 - 5-2 0,32.

5. 3 ,

, (

)?

.

.

k A i , A 2 , . . . , *,

*

,

\ \, m2 2, . . . , -

Ak,

Pn(mh m2 , . . . , m k) = ^ ----- -, - -

\ + m2 +' . . . + mk = . , ,

mi -

, . . . , /*, -

Ak, p f p2 -**- ^ *

\ 1 ,

( \) m2 C%im ,

( \ m2) m3

, . . , -

p f ^2 > 0 -

/^\/^2 WW3 > -\ ^ \ ^ \ 2 ' * * ^ rri\ ^ ^ -2 \ m 2 mk - \

\ ( \)\ ( \ m2)!

m i ! ( n m i)! m 2 ! ( m i m 2)! m 3 ! ( m \ m 2 m 3 )!

( - mi - ... - mk- 2)\

(n - mi - . .. - mk~2 - mk- 1)! n\x

mkl0! mi!m2! .. m^!

.

5. , -

. -

, 365

. *, k = 1,2, . . . , 365, -

,

k-Pi . pk = P{Ak} = 1/365 k = 1, 2 , . . . , 365.

{} , = 730

( )

Ak, = 1 , 2 , . . . , 365, . ,

{5} :

, . ,

{} = 1 {} .

6. -, , -

.

.

( ),

, ,

:

. . 16 5 ,

.

7. , 13 , -

52 ,

. -

(. .

) = 1/ 2.

\ , V 2 irnnne ,

-, _ /2^730 730730 - 730

' 1 ~ 2365 65730

. (..

)

26! 26!

_ Cl6 & _ 21241 11! 15! _ 5 ' 13 13 19 _ n ~ ~ 52! 41-43-47-49

13! 39!

,

, = 1/2

(2) = = 2! ' !>TJ = ^ i r = 0 .0 952 "

8 . ,

, -

?

.

. -

-.2 _ 12!

' 12 2! 10!66.

,

,

. , , -

66.

. Pi ,

1- ; ,

2- ; \2 ,

.

\2 + \ + ,

P = Pl2 - P l - P 2.

^ 66())6,

'-(' --!-)

31-66 31-11

32 6'66 = ^ = ^ = 0 , 00137 . . . .

9. 250 500 . -

, .

. , -

.

, i- -

, = 1/250, / = 1 , 2 , . . . , 500.

-

,

. = 500

(3 < k < 500) = 1 - 5(0) - PsooO) - >(2),

(0) = 500(1 - ) 500, 5(1) = >1(1 - ) 4">

50(2) = |002( 1 - ) 498.

, ( = = 2)

5(0) ^ ~ = ~ 2 = 0,135 . . . ,

5(1) = 2 ~' = 0,270 . . . ,

o o ( 2 ) f e - A = 2- 2 = 0 , 2 7 0 . . . ,

00(3 < k < 500) 1 - 0,135 - 0,270 - 0,270 = 0,325.

10. . - , ,

. ,

, -

( = 0,1,2, . . . ,N\ N

).

. 1 2. -

, 1- , 2-

; 2-

, 1- . -

{ + } = {} + {}.

, {} = {}. , , {}.

,

, (

1/ 2) 1- .

{ = N + (N )

N , ( + 1)-

( + 1)- 1

(

\ n = N -f (N )

N

1 / 1 \ N / 1 \ N / 1 \ 2^ -1

' :0 ,

nN ( \ \ 2N~r _ (2 N - r ) l ( \ \ 2N~r

2N~r \ 2 j N \ ( N - r ) \ \ 2 J

5.

(Q, , ) ,

(. 2). X = () -

, -

>, = {;}. , -

X , ,

,

{>) < , - .

, ,

(. . )

.

, -

{ < }.

F(x) = { < } -

, (..)

X. , F(x)

, lim F(x) = 0, lim F(x) = 1.> -

\ , 2 , -

\ < 2.

{ 1 < < 2} = F { x 2) - F(x i), (1)

. .

(, ) -

.

,

{ < 2} = {X < \} + {\ < X < 2},

{X < \} {xi < X < 2} ,

{ < 2} = { < xi} + { 1 < X < 2},

(1). -

,

-

.

X , -

: \,2 , . . . .

\ *2

\ 2

pi = { = *},

. , ,

P{(X = X l ) + (X = X 2 ) + . . . } = ' i P i = I-

.

.

1. . X -

, -

.

-

0 1 2 k

(1 - ) ! - )"-'-------------------

( 1 - ~ 2 Cknpk(l - p)n~k

\ =

(. 4): -

\ n kP{X = k} = C p*( \ -p )

2. . X -

, -

-

.

1 2 3 k

0 -) ~? (1 - p f - ' p

, , -

.

3. .

,

-

1

- - k\

> 0 , { = k},

k = 0 ,1 ,2 , . . . , (. 4)

{ = k} = - :

,

(, , ), -

: . -

-

()

( , -

), : -

-

.

-

,

f(x),

F(x) = f{t) dt.

, , -

, f(x) -

. -

) Fix)

, +00

fix) dx = 1;

\ 2 (Xi < 2 )

2

(2)

X,

(2) -

. , F(x) -

,

F '{x )= f(x ) .

/() , -

(3) , { = } 0, . . -

-

,

. ,

-

,

. ,

/() ,

{0 < X < + } /() ,

.. /() -

, -

+ . -

/().

.

1. ,

/(*) = { x i abi

(, [ , \ .

, , < . -

[,] . -

, -

[x\,x

2. ,

< ,I0'[^ 1 _*, > 0; , > 0. ,

.

X

: : 0;

3. ,

/(*) = _L_e-(*-*)2/( 22) > 2

, , > 0. -

, . -

X

F (x )= J f ( i ) d t = | at =

{xa)/(T

) -.e -

. F(x),

.1+ 2 + + 4 > 4.

! + 2 + + 4 = 1, F(x) = 1

> 4. . 4.

kF(x)

, ,

(), -

, -

, -

. -

F(x) 1. -

, ; -

.

2. X :

-

X.

X < {,

\

) = < \ + 2

\ + 2 +

\ < X < %

.

+00

J f{t) dt 1,

+

J Ct~V2 d t = 1,1

,

= 2 -

=/ < l.

X

, < 1,

F{x) . > 1.

3. Y = \ / , X -

, 2. -

fy{t) Y {0,1 < < 0,2}.

. Fy(y)

Y.

() = { < } = { < } .

2 , (, 1) -

X . , -

l / -

(, 0) U (1, +).

{ ^ < / } = 0 < 0 , { ^ < / } = 1 > 1.

0 < < 1

> \>

,

FY(y)

0, < ,

/, < < 1,1, > 1-

,

,1

2^

10,

< , < < 1,

> 1.

0,2

{0,1 < Y < 0,2} = | dy = F y {0,2) - F y(0,l)

0,1

4. X, ,

1/4, 1 1

(1/ 2, 1/ 2).

. 1 1 -

1/4 , (1/2,1/2) -

X 1 - (1/4 + 1/4) = 1/2. /

(1/ 2, 1/ 2),

1/2

/ dx

- 1/2

1

2

/ = 1/2. F(x)

X. < 1

F(x) = { < } = 0;

1/2 < < 1/2

F(x) = { < } = { ( = -1 ) + ( ( - 5 . * ) ) } =

1 * j 1 1 / 1 1 1" 4 + J f d x - 4 + 2 \ X + 2 j 2 + 2;

- 1/2

1/2 < < 1

F W = P { X < x } = p { ( X = - l ) + ( ^ e ( - i , I ) ) } = I + I = | ;

X > 1

F(x) = { < } = 1.

F(x) . 5.

. 5

5. , X -

Fx (x), -

Y = Fx(X) -

[0 , 1].

. Fy{y)

Y. , -

Y [0,1]. ,

,

[ 1, > 1-

(0,1].

Fy(y) = P { Y < } = P{Fx (X) < }.

Fx(x) ,

F% 1 Fx(X) < -

X < F% 1 (). ,

P{Fx (X) < } = { < F x \ y ) } = Fx (F- '(y)) =

,

, < 0; () = \> 0 < ^U > 1.

, -

[0 , 1].

6. , - . ,

, ; -

,

; - ,

; . . K(t) -

, -

(0, f]. ,

,

K(t) / > 0 -

/, . .

P{K(t) = } = (Xt)2 , = 0 , 1 , 2 , . . . .

-

. -

.

Z

(. . t = 1).

. F(Z) .

Z

,

( [] ), . . -

F(k), k = 1,2,

F(k) = P{Z < k } = P{(Z = 0) + (Z = 1) + . . . + (Z = k - 1)} =

k - \

= J2 p i z = m}.m=0

,

OO s \

P \ Z m \ P ! 030 (m + 0 I _____^ ~ Z-s ]^ m J

1 f 1 _

(^ ( + /) J | ( + 1) J/=0

= ' ( 1 p)l Xm+le~x - V + p)l Xm+e~x -2 ^ L m+lP I 1 P) + i L m\n p * (/ + /)!/=0 /=0

_ Y ' P*(l - p)lXm+le~x _ Xme~x m A A'(l - p)1~ m i l l ~ m \ P l\

1=0 1=0

,

(-1~\

P{Z = } = * - ! _ *-> = L J m! !

,

k - \ \ m n m - A p

Fik) = P{Z = m i = ^ =/=0 m=0

= e~Xp( l +Ap + (Ap)2 + (Ap)

,*-l

1! 2! (* 1)!

7. X -

f{x). g(y) -

Y = + /3, ,(3 , 0.

. F(x)

X, G(y) -

Y. > 0

G(y) P{Y < } = { + (3 < } = { < - ^ - } =

^ - ^ - ) - ( ) -dy dy

< 0,

G(y) = P { Y < y } = { + (3 < } =

= | > = 1 - | < - ^ \ = 1

,

() = ^ ^ ~ ) -\\ \ a J

8. X

, . .

F(x) = { < } = ~ ^ 10, < 0.

Y = 2.

. G(y)

Y. >

G{y) = P { Y < } = { 2 < } = { - ^ < < ^ } =

= F { j y ) - F i -y / ) = F i ^ y ) = 1 - -'-

d ~^

< 0, , G(y) = 0 g{y) = 0, -

Y .

9. , .

.

1- , . .

. -

, :

, 1/2 ( ,

), -

, 1/3 ( ),

1/4 ( ), .. -

, -

, .

10. F(x) X, Ve > 0 35 > 0, , \F(x\) F(x2)\ < ,

\\ | < 5 \,2 X. F(x) -

-

. ,

< X < +.

. > 0. -

F(x) lim F(x) = 1, lim F(x) = X + + Q O X OO

, > 0, , -

F(x) (, ] [, +)

/2:

', " (, - ] |F{x') - F(x") | < ~,2 (4)

'," [, +) \F(x')- F{x")\ 0 ', " G [-A f,] |F(x') - F{x")\ < | |' - "\ < 8 .

(5) \ 2 (, +).

, \ \ | < 5, \F(X\) F(x2)\ < .

\,2 (, ] \,2 [ ,] \,2 [, +),

. :

1) \ (, - \ , 2 (- , );

2) \ (,), 2 G [1,+).

1)

|F(Xl) - F(x 2)I = |F(x 2) - F(M) + F ( -M ) - F(xy)| =

= 2) - F (-M )) + (F(-M) - F(Xl)).

( X l , X 2 ) , \2 ( )\ < S (5)

F( x 2) - F ( - M ) < - .

(4)

F ( - M ) - F ( Xl) < e- .

,

\F{x x) - F { x 2) \ < - + - = 8 .

2). .

6.

1. . - X ,

X I *2 x k

\ 2 Pk

, \

\, 2, . . . , *

&, \ + 2 + . . . + = . -

X -

(1)

,

> .

, 2 ^

2 ,

Pk,

(1)

\\ + 22 + . . . x kpk. (2)

(2)

X M X {}. X

,

J 2 XiPii= 1

. ,

X .

X -

f{x). -

, X -

[, \. [, ]

\,2 , . . . , -\:

= xq < \ < 2 < . . . < - \ < = .

*k pk = P{xk-1 < X < Xk} = [ f(x) dx, k = 1,2, . . . , . -

~ X k ~ x 3xk G {xk_\,Xk), , pk =/(&)&,

A x k = X k ~ X k - \ .

X ,

X I *2

1 2

X

X ,

, .

X

M X = f ^X i f (x i )A x i .

i= i

>

| xf{x) dx ,

X.

, X

M X = J xf{x) dx,

-

X. ,

, ,

. ,

.

, -

.

1) :

{} = .

2) -

:

{ - } = - {} .

3) -

:

{ { + 2 + . . . + } = { i} + { 2} + . . . + { }.

4) -

:

{ 2 . . . } = { } { 2} . . . { }.

, \ , 2, . . . ,

, Xil9Xi2, - . - ,X ik -

k {^ #i}, {Xi2 2} , . . .

. . . , {Xik g Bk}, B \ 9 2, . . . , Bk - .

5) X -

X i 2

pi 2

() , -

X -

(). Y = () -

M Y = ' p i (Kxi).i

6) X

/(), Y = () -

.

X , -

. MX. -

\ , 2 , . . . , ,

. Xi,

i = 0 1, :

J , - ; = I 1II, - .

Xi, i = 1,2, . . . ,, -

:

0 1

1 -

, i 1,2, . . . , M X 0- (1 ) + 1- = .

, ,

X = \ + 2 + . . . + .

,

M X = MXi + 2 + .. . + = .

X ,

.

M X = Y2 kqk- lp = p ^ k q k- 1, (3)

k = \ k= \

q = 1 - . (3), -

? * -k=l

-

q < 1

b\ = q. ,00

q,

-

:

k = \ k = l k = \

(4)

1 ^ -1k=\

(3)

1 1 1M X = ----------5- = - S = - . (1 ~ q f 2

-

X, . :

00 \ k \ k \ k - i

M X = ^ 2 k J \ e A = Y 1 ( k - l ) \ e X = Xe E (k - 1)!'k = 0 k = l k = \

,

x x} x^ex = \ + - + + +

1! 2! 3!

x ( ).

* 1 ......= { + - + +

(k - 1)! 1! 2! 3!

\*-1 \ \ 2 V ___ = 1 + - + + + = 2 L f ( k - ! 1! 2! ^ 3! ^ k -

\ - \

M X = \ ~ V - = \ ~ = .^ (k - 1)!k = \

-

, 5. X -

\,],

h # + b

= .

X ,

- +

= xf(x) dx =

Xxdx =

, X -

,

+00 +00

= f xf(x) dx = [ x _ ^ e-(x-a)2/V2) dx .J J yfbta

a t,

+

= l J L e- t2/ {22)dt =v 2 i

-

+ +

_ L ^ e-t>/) dt + a _ _ -^/(2

. .

X MX. X

\

\

*2

2

5)

DX = - M X f Pl.

i

X

f(x), 6)

DX = ( - MX) f(x) dx,

-

X.

(6):

{ ( MX)2} = { 2 2 MX + {MX)2} =

= { 2} - 2M X M X + (MX ) 2 = { 2} - (MX)2.

, X

DX = { 2} - (MX ) 2 (7)

-

.

, (6) DX > 0, (7)

{ 2} > ()2,

. .

.

.

1)

:

D {Y + } DY.

2) -

:

3)

:

D{X j + 2 + . . . + } = DX 1 + DX2 + . . . + DXr(.

, , 3).

. :

D{X 1 + 2} = {{ + 2) - ( 1 + 2) } 2 =

= {{ - ) + (2 - 2) } 2 =

= { { - )2 + ( 2 - 2)2 + 2 ( - M X i ) ( * 2 - M X i ) } =

= M { ( X x - M X x ) 2} + { ( 2 - 2)2 } +

+ 2{{ - ){2 - 2)} =

= DXx + DX2 + 2{( - M Xi)(X2 - 2)}.

\ 2

{( 1 - 1){2 - 2)} =

= { 2 - 2 - 2 + 2} =

= 2 - 2 - 2 + 2 = ,

D{X 1 + ^ 2} = DXx 4- DX2, . . .

, 5. X

,

. (. )

= + 2 + . . . + ,

, 2, . . . , -

, :

1 1 . 3)

DX = DXx + D X 2 + . . . + DXn.

DXx = D X 2 = . . . ^ DXn = (0 - ) 2 (1 - ) + (1 - ) 2 - = ( 1 - ) ,

D = tipq,

q = 1 .

X ,

. DX

(7). , M X = 1/. ,

M X 2 = k2qk~lp = a V a _ , ^ k 2Xk* 2 = 2^ = * - ^ ' ()

k = 0 k = 0

'

;k=0

k\

-

:

\ k - \

EkX \

k = 0k\

:

k = 0

-

:

k=0

k X

k\ + ,

, ,

00 2\*S T ~ = \ e x + X2e \

k=0k\

, (8),

2 = _( + 2) = + 2

DX = 2 - {) 2 = + 2 - 2 = .

, DX = .

, -

5. X , -

[, \,

( a + b \ 2 1 ,x dx =

V 2 ) b - a

i b - a )2

12

X, -

,

+00

DX = J ( j c M X ) 2f{x) d x = ( * _ I ) Ae~Xxdx = ^ .0

, X ,

,

+

DX = J ( x - MX) 2f{x) dx i x - a ) 2 ^ =_ e-(^-a)2/(2

-

, -

\ [ . ,

(9)

X.

-

(. 2).

2

X MX DX

:

{ = } = Cknpmqn~m,

= 0 ,1 ,2 , . . . , \ q = 1 \ , : N, [0, 1]

npq

:

{ = } = qm~lp,

= 1,2, . . . , ; q = 1 \ , (0,1]

1

2

:

~

= /! '

0 ,1,2, . . . ; > 0

:

[ [ , 6], /(*) = { ~

[ 0, [,].

, : b >

+

2

- )2

12

:

0, < 0,/(*) = {

\ ~, > 0;

> 0

1

1

3?

:

/ ( * ) = J _/27

, > 0

1

1. -

,

.

.

. X

. , X -

:

* = * 1 + * 2 + 3 + * 4 + 5 + * 6 ,

: ,\ = 1 ; r 2 , -

t% () -

- , ,

;

,

^ to, () -

4 ,

;

,

t$ ^ () -

6 , -

.

, 2 , , 4 ,Xs,Xq -

, ,

: 2 = 5/6, ^ -

= 4/6 , 4 = 3/6, $

= 2/6, Xq = 1/6.

M X = \ + M X 2 + M X 3 + M X 4 + 5 + 6 =

- 1 _+ 5/6 + 476 + 3/6 + 2/6 + 1/6 ~

DX = DX\ + DX2 + DX 3 + DX 4 + DX5 + DX6 =

= 0 4- _ J /6_ + J / ! . + J Z L 4 J Z L 4 J / 6_ = (5/6)2 (4/6)2 (3/6)2 (2/6)2 (1/6)2

2. , , , -

, -

. ,

.

.

. X -

. , , -

X -

\ j . ,

3. 2 ,

.

. - X

-

. X 1,2, . . . , .

. ,

, /, / 1,2, . . . , , , i-

, , i-

.

{ = 2} = />{,2} = {2 ,} { ,} = - J - j . 4-1 = 1,

{ = 3} = {{2 3} = {3 1 2,}{ 2 1 ,}/>{,} =

= - = , DX =

2~ = ( - 1).

{ = 1} = - .

1 2 1 1

2 1

,

{ = } = { = } = . . . = { = } = - .

,

^ 2 _ ( + 1 )(2 + 1 )

5 , :

p i y (az + 1)(2az + 1) ( -f-1 )2 ? I

~ 6 ~ 4 ~ 12~*

4. . , ,

.

.

.

. -

X

X

, ,

M { X \ A } = Y t x iP{X = x i \A},i

( -

).

# # 2 , - - . -

{ = Xi} = /> { * = Xi | Hj} P{Hj).

M X = ' x i P { X = xi}.

i

{ = xf} -

,

M X = > { * = Xi I Hj}P{Hj} =i j= 1

= = * I } = I > m = I =i /=i /=i ;

= p E * p { x = *i\Hj ) = Y / M { x \ H l } p(Hj}.

, \,2, . . . ,

= ^ { } }

/=1

-

.

4. \ , -

, X 2 . -

, .

{ + 2} = M { X i + 2 \ } { } + { + 2 | }{} .

{\ + 2 \ }. ,

= (1 + 2 + 3 + 4 + 5 + 6) i = 7~,

M X2 - MXi = 7~, M{Xi + X 2} = 7.

,

M{Xi + X 2 | A} = 2 i + 4 i + 6 i + 8 i + 10 i + 12 i = 7.

,

7 = M { X x + X 2 \ A } - ^ + 7 - 1- ,

M { X x + X 2 \A} = 7.

5. X , , {0 < X < 1} = 1.

DX < MX.

. DX = M X 2 (MX)2,

DX < M X

M X 2 - (MX) 2 < M X

M X 2 < M X + (MX)2.

X ,

M X 2 = J 2 X1 = X i } < J 2 x i { = x i } = MX,i i

0 < Xi < 1. X -

/(), {0 < X < 1} = 1

f(x) (0,1), -

0 1:

M X 2 2f(x) dx < | xf{x) dx = MX.

, M X 2 < MX, , M X 2 < M X +

+ (MX)2.

6. X \ , X 2 , . . . -

2. Y

\ , 2 , . . . -

, M Y = b, DY = d.

z - ' t x i .i 1

M Z DZ.

. -

. -

W -

f w ( t ) = M { e itw}, (10)

i : i2 = 1. , M{\W\k} <

k,

(10) -

-

-

.

. -

fz( t | k) Z Y = k:

fz(t | k) = M { e itz | Y = k } = M {e it(x+x*++**)} =

= M { e itx' } M { e itXi) .. .M { e itXk} = (fx (t))k,

fx( t) Xj / = 1 , 2 , __ fz(t)

) = / z ( * | k )P{Y = k} = J2(fx(t))kPk,k=\ k=\

pk = P{Y = k}. .

iMZ =dfz(t)

dt t= 0k = \

= t= 0

= J 2 kpkdfxit)

k = \dt

dfxit)

t= dtJ2 kPk = iMX M Y >

*= k=\

J M Z = MX M Y = mb.

DZ = M Z 2 - {MZ)2. 7WZ2:

? 2 = =dt2 t=

= f ; kPh [( - \ ) i fxi t ) ) k - 2 ( ^ ) 2 + ifxit))k- 1^ ^ -

k=\

2 d2fx it)

t= 0

k = \dt /=oJ

= Y , kPb((k - 1W M X ) 2 + i2M X 2) =k=\

= i2M x 2 ^ 2 k p k + iiMX)2 ^ 2 i k - i ) k p k.k=i k=i

M X = m, M X 2 = DX 4- iMX ) 2 = a 2 + m2,

MZOO OO

2 - (a2 + m2) kpk + m2 ]T ( ~ V kPk-k=\ k=\

, kpk = M Y = 6. ^ { k 1 )kpk'-=1 k=\

OO OO OO

J2ik~ l )kpk = 52k2pk-J2kpk =6=1 6=1 k=\

= M Y 2 - M Y .= DY + iM Y ) 2 - M Y = d + b2 - b.

,

M Z 2 = (2 + m2)b + m2{d + 2 - b) = a2b + m2d + 22,

DZ = M Z 2 - MZ)2 == a 2b + m2d.

7. N , ,

, . -

, -

[0, ].

, -

.

. ,

,

Z = \ + 2 + . + /V>

N , , - ,

/- , / = 1,2, . . . ,

[0, ],

Af* = , Det = f 2 .

N - ,

M N = DN = .

, , N,e \ , 2 , . . . -

, 6

:

M Z = { M N = - \ ,

DZ = Dst M N + (Me*)2 DN = ^ + ^ =

8. - . \ , 2 , . , -

, , X * 1 0 -

pk = 1 - , , MXk = pk,

DXk QkPk-

S n = Xi + X 2 + . . . + X n.

S n

pk k -

, k = 1,2, . . . , .

M Sn = pk, DSn = Y QkPk-k=\ k=\

p = (pi + p2 + . . . + pn)/n

. DSn -

,

?

. DSn

DSn = Y ( l ~Pk)pk = Y p k - Y , p I = - 1 - 1)k=\ k=l k=l k=\

n

pk, k = 1,2, . . . , n, , Y P k = np .

k=l , pk .

k=i, :

Y p I -* * Y P k = - k=\ k=\

1 = > 1 + ( > * ~ ) \ k=\

2\ + ,

2Pk + = ,

Y , p k - n p = o,k=\

\ + 2 + + \ = 2 = = Pk =

, , (-

A p k = p k - p )

+ *)2 - f > 2 > ,k=\ k=\

2p A pk + Y , ( A Pk) 2 = 2p J 2 A Pk + 1 2(A P k f = 52 ( A Pk)2-k=l k=l k=\ k=l k=\

n n

, Apk = 0, Y^Pk = const. k=\ k=\

(11). , -

,

S n \ = 2 = = - - , : -

pk, ,

, . , -

-

;

,

.

9. , - (

), ,

S , ( \ S ) me~xs/ \,

= 1 , 2 , . . . . .

R -

. MR - DR.

. F(r)

R.

. R -

, ,

. ,

. .

F(r) = P{R < r } = {} = 1 - {} = 1 - - 2.

, /() R

. dF(r) 2~\ > 0;

= ~ d T = | 0 , < 0 .

- -

MR = /() dr = | 2-7rAr2e 72 dr

+ 0 0 - -

rdfe-***) = (-re-"**) | + j e~*Xr2 d r = | e~r>j3 dr.

t = 2.

MR =

- -

e~t2/2 d t = y/bc 2VA'

1 e ~ t 2 ! 2 d t = 1

/2 / .0 0

:

- 2 + 1DR = M R 2 - ( M R ) 2 = J t 2 f ( r ) d r - \ ^ - \ = | 2 \3~ * 2 d r - - ^ .

= 2,

- ^ -

DR

- -

= J ue-*Xad u - ^ = | u d { -e ~ nXu) - - ^=

-

= ( - ~ ) |+ J e~*Xud u - - ^

1 - | + 0 - - = - - = .7 1 4 7 4 47

10. 1 N. X , ,

. MX.

.

- , = 1,2, . . . , .

X = { ^2, ,}-

N

M X = Y , k P { X = k}.

k= \

{ = k}.

{ < k} = { { 2, . . . , } < k} =

= { < k , X 2 < k , . . . , < 6 } (1).

\ , 2, . . , -

{ < k } P { X 2 < k } . . . P { X n < k } = ( ) ,

{ < k - 1} = ( * ^ 1 ) " .

, X -

,

{ < k } = P { X < k - 1} + { = k},

^ -.

,

MX = j r k P { x = k} = ' k" * ' - * < * - ' r =

k = \ k = \

= ^ ' t { k n + l - ( ( k - l ) + l ) ( k - l ) n) =

k \

= ^ f : ( k n+i- ( k - i ) n+i- ( k - i r ) =

k=\

= i ( x > + ' -

. , Fx ( x )

X

Fx () = { < } = { < }.

< 0 . > 0, :

In*

P{eY < x } = P {Y < \} =

z = (t )/, -

( I n * )/ 0;

. 2 \

fx(x) X

f (v\ dFx(x) - = '

,

,

, . -

, cov(2f, 7)

X 7

cov(*, 7) = - M X ) i Y - MY)}.

M {X Y } M {X }M {Y } . ,

cov(Z, ) = 0,

. , ,

. ,

: ,

, .

,

D { X Y } = DX + D Y 2 c o v ( X ,Y ) . (1)

, D{X } =

= DX + DY. (1) . \,

2 , , ,

,

D{Xx + * 2 + . . . + } = DXi + 2 v(Xi, Yj).i= l i

.

1. (X , ) G, -

:

fix, ), ( , ) G;

I / S q , ( x , y ) e G .

S q G.

2. (X , Y) - , -

:

1fix, ) =

2\/[ - 2X

{ 2(1 - 2)2 (

( - )22

( - ) (

.b) + (jr )

, , , , , > 0, > 0, |r| < 1.

,, ,,

: , X, Y :

= MX, b = MY;

2 .

2 - 1,2,3. -

:

{ 2 = 1} = { 2 = 1 1 * i = 1 } { = 1} +

+ { 2 = l \ X x = 2}P{Xl = 2 } + { 2 = l \ X x = 3}P{*i = 3} =

- + - ~ 14 3 + 14 3 14 ' 3 ~ 3

/ = 1,

\ = 2, = 3. ,

{2 = 2} = , = 3} =

. . 2 -

\:

2 1 2 3

1/3 1/3 1/3

. , -

, , ,

-

Ai 2 , ,

.

Y = X 1X 2 . : 1,2,3,4,6,9. -

. :

P { Y = l } = P { X l = l ,X2 = i} =

= {2 = 1 |Jf, = 1}{, = 1} = 1 . 1 = 1 .

,

^0' = 4 } = | . P{Y = 9} = .

,

P{Y = 2} = { 1 = 1, 2 = 2} + {\ = 2, 2 = 1} =

= {2 = 2 \X i = l}P{Ai = 1} + {2 = 1 1 *1 = 2} { = 2} =

5 1 5 1 5

,

/> = } = , { = 6} =

, Y \

:

Y 1 2 3 4 6 9

2/21 5/21 5/21 2/21 5/21 2/21

:

= 2 = 1 I + 2 1 + 3 = 2,

{ } = 1 . | + 2 . | + 3 . | + 4 . | + 6 . + 9 .

,

1cov(Xh X 2) = { 2} - { } { 2] = - 2 2 =

2. 1

5 . -

. . , 2 , , 4 ,^

. , 2 , , 4 ,$

(. ),

1 2 3

1/3 1/3 1/3

i = 1,2,3,4,5, v{Xi,Xj) = ^ i /.

DXi = MX? - ()2 = | ,

(2)

D{Xx + X 2 + X 3 + X 4 + X 5} = 5 - 1 + 5 - 4 - ( ~ ) = ^ .

2 _ _ 8 3

21 _ 2 1 '

3. X , -

/( ) g(x ) . -

( 2), \ = min{A, },

2 = { , }. -

(Ai,A2). . F(x) = { < }, G(x) = P {Y < }.

f(x) = F'(x), g(x) G'(x). F(x i ,x 2) -

(, 2) :

F(xh 2) = P{Ai < 2 < 2} =

= P{min{A, } < \ , { , } < 2}

= {(( < ) + ( < < x 2)(Y < 2)} =

= {( < \)( < 2)( < 2) + ( < )( < x 2) ( Y < 2)} =

= {( < { 2})( < 2) + ( < min{xi,X2})(A < 2)} =

= /7(min{xi,X2})G(x2) + G{mvci{x\,x2})F{x2) -

{ < min{xi,X2}, < min{xi,X2}, < 2, X < x 2J =

= F(min{x\ ,x2 })G(x2) + G(min{xi,X2})F(x2) -

P{A < min{xi,X2}, < min{xi,X2}} =

_ F (x i)G (x 2) + G (x i)P (x 2) - F(xi)G(xi) x x < x2;

\ F { x 2) G{x2) + G ( x 2) F{ x 2) - F { x 2) G{x2) = F { x 2) G( x 2) x\ > x 2.

f ( x \ ,x2) -

()

~ \ - d2F(*i,x2) _ \ f {x \)g{x2) + g ( x x)f (x2) xi < 2; 1,2) [ 2 1 > 2

V 4./ X , -

- -

. , Z X + Y -

. -

X -

X + .

.

{ = k} = P {Y = k} = ^ ~ ,

k = 0 ,1,2, . . . .

{ + Y = } = { = , Y = } + { = 1, = - 1} + . . .

. . . + { = - 1, Y = 1} + { = , Y = 0} =

/ \k \ m~k \ \k\m-k= V ' f ~ ^ l p - ^ y 1 = - ( * + \ ) V _

-^ \k \ (m k)\ J -^k\(m k)\6=0 4 7 6=0

1 w I= -(*+)_1 ' rn. yk m_k =

ml kl(m - k ) l y

= e-^+A,) 1 ( + A )m = (At + Xy)m e~(\x+\y)ml y ml

. e. X + Y + .

{ = I + = ) - = >f + r - - ) =

y k y m - k

_ { = k, Y = - k} _ t \ e~Xx ' (m - k )\e~

= (, + ,) \_ ( ,+,)ml

I \ k \ m - k , \ ^ / \ \ m k l \ [ Ay ^

!(/ fe)! (; + A y)m m ; + Xy J \; +

. . X - X + Y = - m = /{ + )^.

(j^. X (. . - ) -. X Y , .

.

cov(X, Y) =

X

^ cav(*, Y) = M{XY} - M {X}M {Y} .

{} = 0,

f ix) = - 7=L _ < r* 2/(2^ 2>v 2 tt

X ,

{ +} = [ dx.J V 27;

= 2&, { +1} = 0.

= 2k I.

{ +} = [ L - x 2ke - x2/ (2a^ dx,J v 2

, t = / 1

{ +} = J - = < jf t2ke - t

. .

I2k = (2k - 1)/2*_2 = (2 - l)(2k - 3)/2fc-4 =

. . . = ( 2 k - l)(2k - 3) . . . (2 k - (2 k - l))/0 =

= (2 k - 1)!!

+oo

1 _-* I2d t = (2 k - 1)!!.

, n = 2k \

M {X n+x} = M {X2k} = a f ( 2 k - 1)!!. (3)

,

a 2 = D { Y } = D{Xn} = M {X2n} - (M{Xn})2.

n = 2k 1 a 2 = M {X4k~2}, -

(3)

a 2 = a4k 2(Ak 3)!!, ay = a2k ly/(4k 3)!!.

, n = 2k 1 (. e. ti)

a f ( 2k - 1)!! _ (2k - 1)!!r x y =

/(4* - 3)!!'

3

V l5 VT5

/7

, = 1, = 1; = 3, =

= 5, *,, = - ^ = . = 2k (.. / ),

= 0 .

6. , - .

.

X Y.

D{X + Y} = DX + DY + 2rVDX\fDY,

TO

D{X+Y} Vd x Vd ?

V M V d y Vd y \[d x

t = . Vd y

1 / 1\

~ 2 \ 7 / w > 0 -

, > 1, minf i + l ') = 2./> \ /

,

D{X Y} = DX + DY 2 r V M V D Y ,

D { X - Y } _ s/ | s /B Y 2 r

s/b x V b y s/b y s/d x

t = y = , TO s /B Y

W > 0 t + - 2 r > 0 ,

r < - 2 (t + -t ) v o o .

, < 1. , \r\ < 1.

( j ( . X Y , -

[\,\\ [02,^ 2] -.

Z = X + Y .

. :

fz(z) Z = X + Y, X Y -

fx(x) f Y{y) . -

X ^

X , 6

. -

Z

Fz { z ) = P { X + Y < z} P {X + Y < z } = J 2 P { X = x h Y < z - x t} =i

= J 2 p { X = Xi}P{Y < z - X i } = J 2 f x ( x i ) A x i F Y ( z - X i ) .i i

\ f x { x )F Y{ z - x)dx,

(, ) X. - X +,

+00

F z ( z )= [ fx (x)FY( z - x ) d x , ^ ^!

&, z,

+

f z { z )= J f x ( x ) f y ( z - x)dx.

, X Y :

,

^ 0,2 < z X < 2 ,fy (z ) = { b2 - a 2

0 z $ [2, &]-

(4). .2 < z < &2

Z b2 < < Z 0,2-

z ( (4) ) (4) .

1------------------1---- 1-----------------------1---------------------------- X'Z - 2 Z -

a i + c i 2 < z < a 2 + bi, z < a \ + b2

( \ \ > &2 0,2 , z

\ + 02 < z < \ + 62).

-2

\ 2 2

1 , Z \ 0,2 =

(bi -ci)(&2 - e 2)'1

ai

Z - 62 z - 2

. .

z - 2 < b\, z - 2 > \,

\ + ?2 < z < 2 + \, fz(z)

-

fz(z)1 1

\ \ 2 2dx

1\ \

2-2

\

z - b 2 z - 2

. .

z - a 2 >b\, z - b 2 a \ - ), fz(z)

fz(z) =

Zb2

,

1bi - \ 2 2

1

1 d x = ' + 2~ (bi a\){b2 - 2)"

z b2 z 2

. . z 2 > \, z > b\ + 62. fz{z) = 0 .

,

'0

z \ ,2

{\ - )(2 - 2)

1

\ \

bi + b2 - z

(\ - \)(2 - 2)

0

z < \ + 2,

CL\ +0,2 < Z < \ + 2.

a\ + b2 < z < 2 + &

CL2 + b \ < < \ + 2 ,

> \ + 2 .

(. 6).

. 6

8.^ (X , Y) -

fix, ) =\( + ) 0 < < 1, 0 < < 1;

.

{ + Y > 1}, MX, DX . 7

. -

:

+ -f-OO 1 1

1 = J | f ( x , y ) d x d y = ^ dy^(x + y ) d x =

1

=

. ((+yx)Vyy=A\ih+y)iy=Ai$

+ = ,

. . = 1. (. . 7),

{ + > 1} = | | f(x, ) dx dy = | dx | (x + y) dy =

l l

+

y=i

y = l - x

dx =

ix

x + 1 - x(l - x) - ^ ^ ) dx =

M X DX, /() -

X:

+00

f x ( x ) = | f ( x , y )d y =

(x + y )d y = x + ^ [ 0 , 1 ] ;

^ [0,1].

,

+00

= xfx(x) dx = x(x+i)dx = h

+

DX = M X 2 - (MX) 2 = I x 2f x ( x ) d x - ( M X ) 2 =

oo

2(*+0 ~ 2=2 jn_

144'

, M Y = MX, DY DX.

CM |C

O

- cov(X,Y) = ------------ .

^ X

cov(X, Y) = M{XY} - M { X } M { Y },

-)- -j-oo 1 1

M{XY} = | x y f ( x , y ) d x d y = dx xy(x + y )dy =

4 + 4y= 1

y=00

,

_1_

IT'

9. X, Y -

1{ ) = { ~ - i y2 - ~ * ~ 1% - f )

,, -

(.. -

) :

= 1, = 1 1.

' 2 = 1,

2 13,

A m i Bmi + =25

- " ' ' 2

,

,.2

1 = 2.

1 / ?

= - 1 = 7 0 = v 7cTy a t I

1

2(1- 2),2 2

(1- )

. F(y)

Y. , {0 < Y < 1} = 1

F{y) =1 > 1;

0 < 0.

(0,1).

F{y) = P { Y < } = p { j ^ Y 2 < } = < &1 + 2)} =

= 1 - ) < 2} = { 2 > ^ * i } .

f{x \ ,x2)

\ , 2. , ,

f ( xh x 2) =1, \ [0,1], [0,1];

0, .

(0,1/2]. (1 - ) / + 1 (. . 8)

[ 2 > = ^ f ( x b x 2) d x xdx2 =G

/(1-) 1 y/Q-v)= J dxi | f ( xu x 2) d x 2 = ( l - - -x^j dxi =

V 2 ) Li=

Xl(l-y)/y

2 \ \ X \ = y / ( \ - y )

=0 = 2JT T Y

. 8

[1/2,1). (1 ) / -

1 0 (. . 9)

1 1

|^ 2 > - \ ^ = ^ f ( X \ , X 2 ) dX\ dX 2 = ^ d X \ J f ( x i , x 2) d x 2 = ^

= V i - l z x ^ iXi = ( X l - ^ A Y ' - ' = = J \ 2 / 1 = 0 2 2

,

0 < 0;

0 < < ,

\ < < 1;

F(y) = 1 - = .

M X = , DX = . k2 M X =

= M X k \ .

P { h < X < k 2} = P{MX - e < X < M X + e} =

= { \ - \ < } > \ - ^ = \ --------- ^11 ' 2 - ) /2

. .

/>{*, < X < *2} > 1 - , . - \)

.

) = 1/2, = 100, k\ = 40, k 1 -4 100 i i .

4 (60 - 40)2

) 1/4, = 800, &i = 150, &2 = 250:

/>{150 < X < 250} > 14 ' 8 0 0 - I | _ 4 7(250 150)2 50*

2. , X , {} ( > 0 ), > 0

. -

, f{x).

+ 00 +00 +

{} = eaxf(x) dx > eaxf(x) dx > easf{x) dx =

+00

f{x) dx = eaeP{X > e.},

. e.

M {eaX} > eaeP{X >},

.

3. {) > 0 . ,

{(\ \)},

{ { \ - \ ) } { \ - \ > } [ (\t\)f(t + M X ) d t + f (\t\)f(t + M X ) d t >

{ () > 0}

+00

> | (e)f(t + M X )d t + | (e)f(t + M X)d t =

{ = t + MX}

M X -(-

= (e)f(x)dx+ 4>{e)f{x)dx =

- +

+

= () ^ f{x) dx + f(x) d x j =

OO A4X-\-

= ()(P{X < M X - e } + P { X > MX+e}) = {){\-\ > },

. . ,

{(\ - M X |)} > (){\ - M X | > },

.

4. \,

2 , . . . , , . . . :

0

1/2" 1 - 1/2"-1 1/2"

> 0 . ,

.

. , = 0 =

= 1 , 2 , . . . . , -

, , DXn < = 1,2, . . . .

DXn = MX* - () 2 = MX* = ^ 2.

2/2 "-1 , = 1 ,2 , -

f(x) = 2/2~, > 1.

[1, -|-) +:

lim f{x) = 0.X ++

, [1,+), . . -

, , [1,+) /() < .

2DXn = 2 ) = 1,2, . . . ,

?.

, .

/ 5. , ,

-

, 3/4

780.

. X t , -

- , i 1,2, . . . , .

Xi

Xi 1 2 3 4 5 6

p 1/6 1/6 1/6 1/6 1/6 1/6

MXi = 7- , DXi =

{ + 2 + . . . + } = 1 + 2 + . . . + = - ^ ,

X i ,X 2, . . . ,

D{Xx + X 2 + . . . + X n} = D X l + D X 2 + . . . + D X n = n - ^ .

-

+00

J X l + . . . + X . - M {X l + . . . + X , } ^ 1 _,V2I y/D{X, + . . . + X } I I -Jb :

A

+ o o

P\ X\ + . . + X n > A \ j n + e tZ/2d t

= 780,

< +

[ 4 = e - ^ d tJ V2* ~ 4'

I

- f

3 1

0,675, ,

( ): 228.

6. , , 0,99, 1000-

.

. .

. -

, i- , i = 1,2, . . . ,.

(. 5)

S n = \ + 2 + . . . + .

MXi = DXi = g ,

7 { + 2 + . . . + } = ' - , D{Xx + 2 + . . . + } = ^ .

{|5 - 5 | < } > 1 -DS,*

'

1 - ^ = 0,99.

VDS, 1000= 540 ,0 6 ... .

7 M S n = 2 00 = 3500,

/>{3500 - 540 < S n < 3500 + 540} > 0,99,

. , -

.

f i -ll

MSn

< e l L=e~t2/2dt = 2(),J J 27

Z

() = = e ~ t2/2dt . 2() = 0,99J 27

: 2,58,

^ D S n . = . 00 2,58 = 139,33

{3500 - 140 < S n < 3500 + 140} > 0,99.

, -

S n, . .

.

7. 500 50 . -

, -

. :

) , ) ,

) .

.

. -

,

. . = 50

, -

. , -

= 1/500. X

.

{ > 3} = 1 - { = 0} - { = 1} - { = 2}.

{ > 3} = 1 - -0,1 ^ = 0 ,0 0 0 1 5 5 ....

, . . -

. -

-, - :

{ > 3} = {3 < < 50} = ( f 0 ~ nP ~ =\ \ / (1 - p)J \ / ( \ - p)J

,

P{X = k} = Cknpk{ \ - p ) n- \

:

{ > 3} = 1 - { = 0} - { = 1} - { = 2} =

= 0 ,0 0 0 1 4 6 ....

8. . k .

,

.

. ,

-

, .

X ,

, -

( ) \ / k ( -

):

= 1,2, . . . , .

. ^ . / 1 \ / ( \ -

{ < < } = - j ( i ) ( l - I )

M X = 7 , DX = ^ N*} < (3,

1 - {0 < X < N*} < (3,

{0 < < N*} > 1 - 13.

N*, ,

N ,

-. -

(30 - n)Nat V nN \ V" nN ( , \

= ' = { ~ ) =

_ V ~ n)N

2

% 30

/ V / A r * ! >1*~\ / 0 - MX { < X < N } ( -

/ V

-

< 1 -;2

/2

Pi - S ~ \ F=e 1 1* dt,

I Ox J J V2tt

() -

2

1 -t*

y/bie 1 ^ dt . 2(

9. ,

1. ,

. , ,

, -

, . ,

, -

Q; ,

,

0. 0 -

. ,

. :

1) ,

, ;

2) , , +

U ,

;

3) , ,

, .

, , + , , -

. , ,

F q , . . , -

Q

(, , ,

, T q , , , + , F q ) .

, , , -

:

+ = = , + = + , ( + ) + = + ( + ),

, () = (), ( + ) = + , + = ,

= 0, + = , = , + 0 = , 0 = 0.

', ' -

VI'. -

', ', '+ ' , ' ' . (-

) ,

-

, , , . . . . -

' ,

VI' -

7, ' + ', ''.

, :

, -

, VI'

Tq -,

' T'Q -

, -

' , ',

', +

' + ' , ''.

,

(. . -

, ),

-

Vt' , ,

,

0.

.

, -

, , , . . . . -

, -

, VI ,

, . .

, -

, , -

{},

. , {}

, ,

< {} < 1, {0} = 0, P0 {Vl} = 1

: = \ + . . .

. . . + , At , i = 1 , . . . , ( , G ), A/Ak = 0 / k,

{} = 0 {1} + . . . + 0 {}:-

{}

, . . ,

.

At, ^ , . . .

- 1+ 2 + . . .

\ - . . , .. -

, (

), ,

, . .

. -

a -, ( -

) ;

- . ,

-, - -

\ , ,

.

,

{},

, {}, -

- , '-

, {} = Pq{A}\ {} -

{}, . ., , 0 < {} < 1

= \ + 2 + - - -, ; (/ = 0

i /, {} {\} + { 2} + . . . . {} -

, ,

.

, -

-

, , ,

. ,

, -

.

, - -

{}, , -

, (, , ). .

2. X = () , -

> . ,

X , ,

,

(>) < , - .

X Y ,

( ) , , () = Y(u), -

1.

X = () , F{x) =

= { < } -

,

> 1 +. F() -

(. .) X.

f(x), ,

F(x) = /(/) dt,

f{x) -

X. Xi, . . . ,

-

F{x\, . . . , ) = { 1 < \ , . . . , < },

-

( ) f{x\, . . . , ), ,

^1 %

F(x i , . . . , x ) = | . . . | f ( tu . . . , t n)dti . . . , d t n.

.

- -

, X{t)

t , -

, , -

t. X(t)

, -

(, , ).

t -

X(t), -

t. , t :

t (,

), .

-

. {l,T,P) -

.

-

X(t ,u) , t e

, . . .

, ,

: X(t) Xt X{t,u) .

X(t) .

t = t\ X{t\) X{t\,ui) -

, F(x\, = {(^) < \}.

ti, . . . , tn t.

X{t{), . . . ,X(tn)

F(xuti-,X2 , t2-, . . . ; x n, tn) =P{X(t i) < \, . . . , X ( t n) < }.

= 1,2, . . .

tj -

X(t). -

, -

.

X(t) Y(t) -

, t

t X(t) Y(t)

, . . t P{X(t) = Y(t)} = 1.

-

.

, X(t,w)

t t . , uj -

-

t. -

x(t), ,

, ;

.

x(t) , -

, X(t).

,

. -

, it ,

.

1. , -

.

. it -

, . . P{it = nt} 1 t.

Fz(xh t i ;x2 , h ) = P{i tx < < x2}>

Fv(xi,ti-,X2 , t2) = {

2. {} = 1,

{} = {}.

1, 2 ,

& . n-

= 3 , 4 , . . . .

2. X(t) -

F(xu ti;x2 , t2 ', \xn, tn) V = l,2, . . . W i,*2. -

p * = P { ( ai < X { h ) < a 2)r\[{bx < X ( t 2) h > 2> 2^)

* = {[(\ < U < -2

\ < V < %

\ < U <

\ < V <

=

{ (

* = \ < V < &2

Z > 2

> +< +

(. . ),

 c 2

1- 3- :

( \ c 2

' \ c 2

F\i{u,v)

\ "-1-00

U V.

| ai < U < a 2

\ < V < 2 }:

[ < 2 \ = { < { U < \ \

\ V < b 2] \ V < b x] \ b x < V < 2]+

+

j u < a i \

1 v < b 2) \

{U < a 2| _ | U <

U < \

V < \ \ 1 \ b i < V < b 2

< U < a2) ( < U < 2

V < ) + \ b i < V < b 2

/ < a i ) ( ( U < \

V < 2 )

a \ < U < 2

V

, ,

{a \ = F\\\{a2 , b2 , 2) + Fm(ai, b\,C2) -

Z < c 2 )

- F\u(a\,b2 , C2) - ^ 111(^ 2 > h , C2).

:

* = [^11(^2. 62) - F\w{a2 ,b2 ,C2)\ + [F\\{a\,b\) - F\\\{a\,b\,C2)] -

- [F\$a\, 62) - FnKai, 62. ^2)] - [^, b$ - F\\\{a2 , &i, 2)] +

+ F\n{a2 , b 2 , c \) + F\ \ \{a\ ,bi ,c\ ) - F m (ai, 62, o ) F\i\(a,2 , b \ , c \ ) +

+J , \ &2I

\ \ '2 /

3. -

[0,1]. -

[0 , 1], -

. = {}

[0,1]. ,

,

[0,1].

{0,,,) j(w), t G [0,1], -

:

= 1 t > .

: ) ) < x } =

x > 1: 1 t,

< 0: 0 t,

G (0,1]: { < t} t.

> 1

. 10

F(xu t\;x2 , t2) = {&! < \ ,&2 < 2} =

= P W ' 6 i M < 6 2 < *2} =

" \ > 1, 2 > 1: 1 t\, t2;

\ < 0 2 < 0: 0 t\, t2;

\ > 1, 2 (0; 1]: {> < t2} = t2 t\,

2 > 1, \ (0; 1]: { < h} = t\ t2\

^ 2 (0; 1], \ (0; 1]: {> < m in(fi, 2^)} = m in (^ , t2).

4. rj

F(x). -

& = ] + t.

. :

F^(x, t) = < } = P{q + t < } = {] < t} = F(x t).

:

F^(xh ti;x2 , t2) = {&1 < * 6 2 < * 2} =

= { + h < x i , T ] + t2 < 2} = { < \ - t \ , 7) < 2 - t2} =

= {) < min(Xi - ti)\ = F f m in ^ - /,)).=1,2 i=l,2

, - :

Ft(xh ti;x2 , t2; . . . - ,xn, t n) = P{tl < * & 2 < 2, } =

= F( min ( X i - U ) ) .1= 1,2, ...,

5. & , -

($1, , ) .

.

,

. . *, -

sup & < . , .

.

:

{ ^ } -

, -

,

, , , -

.

(, , ) , = = = [0,1], [0,1]. -

&(w), t [0,1], ), t [0,1], :

& , -

. ,

, .

{: ^(;) } J-.

= const.

= {6 : sup ^ < }, te[ 0,1]

&() 0, t ,

t = .

{) . 11.

k Vt

t

. 11

10. ,

X = (>) -

(, , ) , -

, . . .

- ,

(>) = (. ^ , { /} , i

X {}:

M{X} = J2 xiP{At}.i

X ,

/(),

{ } = | x f{x )dx

, .

( , X -

) ,

.

X(t) -

m(t) = M{X{t)}.

X(t),

, -

t. m(t) -

,

.

, , -

t -

x(t) m(t). -

t D(t)

X(t) ,

t.

\, . . . , ,

F(x . . . , ) = Fi (xi ) . . .F n(xn)

( f(x 1, . . . , ) )

f ( xh . . . , xn ) = f i ( x i ) . . . f n(xn);

Fi{xi) , /;(/)

Xi, i = l , . . . , n .

-

, . . -

X(^i) X(t2),

K(tu t2) = M{(X(t\) - m ( h ) ) ( m ) - m(t2))},

X(t).

X(t )=X( t ) -m( t ) ,

K(t\, t2)

m , t 2) = M { X ( h ) X ( t 2)}.

, K(t, t) = D(t), . .

, , .

D(t) 0, K(t\ , t2) -

m , t 2)k(th t2) =

a(h)a(t2V

a(t\) = ^ D ( h ) , cr(t2) = y/D(t2) -

. k(t \ , t2) -

, k(ti, t2) . ,

, |k(t\, t2)\ < 1 t\ n t 2.

, -

Y h Z M{YZ) = M{Y}M{Z} , ,

k{t\, t 0 , a U V -

-

D. -

m(t) K{t\ , t2) .

. t

m(t) = cos9tM{U} + sm9tM{V} 0,

K{t\, ti) = M { ( U cos9t\ + V sin0i)(7cos0^ 2 + sin#^)} =

= cos 9t\ cos 9 t2M{ U2 } + sin 9t\ sin 9 t2M { V'2} +

+ (cos 9t\ sin 0^ 2 + sin9t\ cos9 t2)M{UV}.

U V

M{UV} = M{U} M {V } = 0,

,

K(t\, t2) = Dcos9t\ cos0^ 2 + Dsin9t\ s in 0^ 2 = Dcos9(t\ ^)-

2. 1, -

U V, 9, :

X(t) = U cos &t + V sin 0^,

U, V, 0 , M{U} =

= = {} = 0, D = M { U 2} M { V 2}, 0

f(9). , -

m(t)

K(ty, 2^) . .

m(t) = M {U cos^} + M { y sin 0 ^ } .

U , V 0 ,

m(t) = Af{I/}Af{cos0*} +M {y}M {sin@ 0 = 0.

K{t\ ,t2), , -

1 X(t)

0

K(tu t2) = M{X(h)X(t2) | = } = Dcosflfa - t2).

K(t\, t2) -

f (6 ) -

,

h) D f{6 ) cos 6 {t\ t2) d0 .

,

= < 2 1

> 0,7 2 + 2 *

0 < 0,

(

| f(O)d0 = 1 ).

m , t 2) = COS.2(V 2) d0 = D e M - M h - t2 1}(2). -f-

0

,

K(t\ , t2) \t\

, .

. /(0)

W K(t\, ^) , - . , , 2) 3) .

(2) - (., : .., .., - .. . .: , 1982).

2/ ( 7) = 0.

. ,

, -

X(t) ( -

).

.

3. , a t\, t2 , . . .

t, , :

1) () , -

, = 0 , 1 , 2 , . . . ,

= ^ {- 7}

;

2) , -

, .

, t\, t2 ,

. -

X(t), -

:

< t < t\,XI t \ < t < t2,*2 t2 < t < *3. h < t < t4,

X1.X2.X3, . . . -

\ , 2 , , . . . -

, , -

D. X(t).

. , m(t) = 0, {\} = { 2} =

= { } = . . . = 0. K(t, t'):

K{t,t') = M{X{t)X(t ')} =

= M{X(t)X(t' ) I A }P {A } + } , (1)

, (t , t')

t\, /2, h , , (t , t ')

t\, t2 , h , . . . . -

, 2, $, . . . M{XiXj}=M{Xi}M{Xj} = 0

i /,

M{X{t )X{t ' ) \A} = 0.

X(t) X(t') , -

,

M{X(t)X(t ' ) | } = 1{2(0} = D.

{} () \t t'\,

K(t, t') = D exp{\ \ t t'\}.

,

.

:

2 , 3 .

,

, -

.

4. X(t) .

-

. X(t)

, -

V -

. V -

Dy.

, X(t).

. X(t)

V.

)} = M {V ) = m v , D{X(t)} = D { V } = Dy.

t' > t, = t' t. ,

t u t '

, = , -

. :

K ( t , t ' )= M{X(t)X{t' ) I A }P {A } + M{X{t)X{t' ) | B}P{B} =

= 0 P{A} + M { i( / ) i ( 0 } ~ = D y e - W - ^ .

t t'

K(t,t ') = - ^ - ('\.

5. X(t)

Y{t)

1 X(t) > 0,

0 X(t) = 0,

-1 X(t) < 0.

X(t), -

.

.

X(t) ,

-

V -

. V /().

Y(t),

, Dy -

Ky(t\, t%).

. t Y(t) -

: 1+

J f{v)dv 1

_ 0 = | f ( v) dv = 1 . ,

m Y = M{Y(t)} = \ - p + ( - \ ) - p = p - p ,

Dy = D{Y(t)} = (1 - ( - p ) f p + (-1 - ( - ))2 = 4 pp.

, t\ ^ -

, =

, . -

:

Y(t, t') = M{Y(h) \A }P{A} + M{Y(t\) Y{t2) I

, Y(t\) (^) -

( )

, M{Y{t\) (^) | ^4} = 0. -

, Y(t\) (^)

, M{Y{t \) F (^) \} = Dy 4pp.

Ky(tu t2) = 0 {} + Dy {} = Dy ~l'1-fel = 4~^ - ^ .

6. X(t) . -

. X(t) -

1 1. -

K(t\ , t2 ).

. , t M{X{t)} = (1) 1/2 + 1-1/2 = 0

X(t\)X{t2) -

: 1

P{X{tx)X{t2) = -1} =

t\ t2

1 {1 - ~2

2

= |Ti ^|> 1

^ (2)! 2=0

,

K{t), , -

[0, t).

,

.

1. : Ptj+T(k) -

[t, t + ) t:

V t, ^G [0, + 00) Pt't+T(k) = PT{k), = 0 , 1 , 2 , . . . ,

. . Pt,t+T(k)

[t, t + ) -

, t.

2. : Pt,t+T(k) -

[t , t + )

, . -

, -

[t , t + )

t

. , ,

-

[^, t\ + t i ) , . . . , [tn, tn + ) \, . . . , -

.

3. : -

t

t t > 0:

lim PW > > ' >=

, 10 3. -

, K ( t ) t ,

At.

. -

M { K ( t ) } = At ,

- .

1. K ( t ) . -

ty < t2 < < tn -

k y < k 2 < . . . < k n

P { K ( t y ) = ky, K ( t 2) = k 2 , . . , K ( t n) = k n} .

. K ( t )

: K ( t ' , t") [/', t").

= 2. , -

P { K ( t y ) = k h K ( t 2) = k 2 } = P { K ( t \ ) = k u K ( t u t2) = k 2 - k y } =

= P { K ( t y ) = k y } P { K ( t u t2) = k 2 - k y } =

= P { K ( t y ) = * , } P { K ( t 2 - ty) = k 2 - k y } =

_ W l)kl \ t I (A(/2 - ty))h ~ k' X(t9 - h )

ky\ ' (k2 - k y ) \

,

P { K ( t y ) = ky, K ( t 2) = k 2 , . . . , K ( t n) = k n}

= C ~ X U . (A(^2 - ty))k2~ k' M U - U ) (A(^3 - t2) )k3~k2 - M t i - U )

ky\ (k2 - k y ) \ ( h - k 2)\

. (A(^n-l - tn- 2) )k"- '~kn~2 -(*,.,-_) x

(kn-y - k n- 2)\

x ( ( ^ ~ t n - i ) ) ka kn~ l c - \ ( t n - t _ \ ) _

(kn - *_,)!

Xkne ~ xtat^'(t2 - ty)k2~kl . . . ( t n - tn- \ ) kn~ kn-'

~ ky\(k2 - k y ) \ . . . ( k n -kn-{)\

2. K(t) ,

a K(t) , K(t)

( ) . ^-

K(t).

. :

t; \ t

; . . . ; t -

;

P{K(t) = k} = p { j r {K(t) = k + m}{Hm}} =

m=

= Y : P { K ( t ) = k + m,Hm} =

{ I K(t) = k + } P{K(t) = k + ) =

m\k\ (k + m)\=

- ) Y , = ^ - " ( 1 -k\ ' \ k\

= 0

_ (A ( l - p ) t ) k _ (1- )k\

. . K(t) (1 ).

3. K ^ t i , h )

K(t) .

. t2 > t\.

Kps(t i ,t2) = { ( - xh){K(t2) - a^2)} =

= M{K(h)K( t2)} - Xt2 M{K{h)} - Xti M{K(t2)} + x 2 t{t2 =

= M{K(ti)(K(t2) - K(ti) + K(tx))} - Xt2 Xti - Xti Xt2 + X2 txt2 =

= M{K(t\){K{t2) - ) } + *{( )2} - X \ t 2. (1)

-

K{t\) K(t2) K{t\) ,

M{K(ti)(K(t2) - K ( t i ) ) } = M { K (h ) } M { (K ( t2) - K ( t x))} =

= Xtt (Xto Ati) X2UU A2/?:

,

M{{K{h))2} = D{K(h)} + ( } ) 2 = /1 + (A*i)2.

(1),

^ps(^b 2^) Afi.

t2 < h

Kpsih, h) ^ 2.

t\ t2

Kpsih, h) = Amin(^b ^)-

, ,

, . ,

t2 > t\

Aps(^i, t2) = M { k { h ) k { t 2) } = M { k m h t 2) - h h ) + k m } =

= M { k ( h ) 2) - m m + M { k { h ) } 2 =

= M { k { t x)} M { k ( t2) - k ( h ) } + D {k( tx)} = - + = x t h

4. ,

, . . -

, -

( ).

, N ,

.

. \ -

, -

1- ; 2

1-

2- . .

{\ + 2 + . . . + > N}.

F(t)

= \ + 2 + . . . + .

F(t) = P {T i + T2 + . . . + Tn < t } =

= { [0, t) > } =

= 1 { [0, t) < } =

_ v (/|" ' 'i = ' . - 0 I- ,0

+oo

3e xt dt

NN n - l e - \ N ( n _ ^ N n - 2 e - X N f a _ _ 2 )N n - 3 e - \ N

\ 1 >0 . Q + . . .

(n - 1 )(n - 2) .. . (n - (n - 2))N

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