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7/26/2019 Giao trinh xac suat va thong ke Toan 2013.doc
1/105
ThS Phng Duy Quang
Gio trnh
XC SUT
V THNG K TON
H ni, h!ng "#$%&$
1
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'(C )(C
)*+ N+ -.U///////////////////////////////////////////////////////////////////////////////////////////////////////////////
Ph0n +/ )1 THU23T XC SUT///////////////////////////////////////////////////////////////////////////////
Ch45ng +6 7i8n 9: ; nh ?!9 @uA///////////////////////////////////////////////////////////////////// 1. Gii tch t hp........................................................................................................
2. Bin c ngu nhin v quan h gia cc bin c ngu nhin.....................................
. !"nh ngh#a $c %u&t c'a bin c................................................................................
(. )c c*ng th+c tnh $c %u&t.....................................................................................
Ch45ng $/ 7i8n ngBu nhin ; uy EuF h5;ng php 5?c 5ng i@.................................................................................
2. >h5;ng php 5?c 5ng :hAng tin c/-..................................................................
Ch45ng R/ Ki nh gi huy8 h:ng W/////////////////////////////////////////////////////////////////
1. Gi thu-t thng :.................................................................................................
2. i@ "nh gi thu-t v6 :C vDng tAn.....................................................................
. i@ "nh gi thit v6 ph5;ng %ai c'a bin ngu nhin ph0n phi chuEn...............
(. i@ "nh gi thit v6 tha % p c'a bin ngu nhin ph0n phi FpH...................
7. Iu- 8iJn thng :....................................................................................................
T+ )+U THY' KHZO///////////////////////////////////////////////////////////////////////////////////////////
2
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)*+ N+ -.U
Gi![ \]nh ?!9 @uA ; h:ng W [!n^ 5c bin %AKn chA %inh vin cc ngnh
:inh tL qun t4" :inh 8Aanh v ti chnh ng0n hng c'a
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Chng ?. 3i@< Anh gi !h&,! !hng D*
GiA t4Rnh Sn Su 4a et bKn Dc nn :h*ng th@ t4nh cc %ai %_t.
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Ph0n +/ )1 THU23T XC SUT
Ch45ng +6 7i8n 9: ; nh ?!9 @uA
_ &/ Gii >9h ` hF1F2 FnV1H l thR
>F1F2 FnH j >F1H. >F2F1H >FnF1F2 FnV1H
Hj u /Nu cc bin c F1L F2L L Fn=c /p tAn phSn thR ta c_
>F1F2 FnH j >F1H. >F2H >FnH
V> de /Q=t h=p Zng } bi $anh v { bi z. X&- ngu nhin 1 vin bi %au _ &- tip
vin bi th+ 2. FH.>B FH .17 1( 17
= = = .
/ Cng h9 9ng ?!9 @uA
)hA F v B hai bin c t4Ang =t php thT. F BH j >FH >BH >FBH
Hj u &/Nu F v B hai bin c $ung :hec thR >F BH j >FH >BH
F >F H= =
=
Nu F1L F2L L Fn h S- ' n bin c thRn
i
i 1
>F H 1=
= .
1o
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Hj u $/ >FH 1 >FH=
V> de L/Q=t c*ng nh0n +ng -L bit cc - hAKt =ng =c /p v?i nhauL $c
%u&t @ t4Ang thMi gian < - 1L 2L :h*ng b" hzng Sn 5t lL lL} v lL{. F2H j lL} >FH j lL{. mc %u&t t5;ng +ng @ \i - b" hzng t4Ang thMi gian < d
1 2 >F H lL1>F H lL2>F H lL= = = .
~A cc - hAKt =ng =c /p v?i nhau nn cc bin c 1 2 F LF L F =c /p tAn phSn.
/- $c %u&t @ c ba - b" hzng t4Ang thMi gian < d
( ) ( ) ( ) ( )1 2 1 2 > F F F > F > F > F lL1.lL2.lL lLllo= = =V> de R/ai $K th' \i ng5Mi ben =t pht vA bia. mc %u&t t4[ng ch c'a ng5Mi
th+ nh&t lL{ c'a ng5Mi th+ hai lL}. FH >BH >FBH
R FL B =c /p nn >FBH j >FH.>BH j lL{. lL} j lL7o.
/- >F BH j lL{ lL} lL7o j lL(.V> de o/Q=t %n phE $u&t $5yng phi qua Sn :i@ t4a. mc %u&t @ =t ph
phE b" AKi y Sn :i@ t4a Su b]ng lL} nu y Sn :i@ t4a Su :h*ng b" AKi thR $c
%u&t b" AKi y Sn :i@ t4a th+ hai lL t5;ng tZ nu Sn th+ hai n_ cng :h*ng b"
AKi thR $c %u&t n_ b" AKi y Sn :i@ t4a th+ ba lL7.
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j 1 lL2.lL1.lLl7 j lL
L/ Cng h9 ?!9 @uA 0y ; 9ng h9 7ay@
a/ Cng h9 ?!9 @uA 0y
)hA B1L B2L L Bn h S- ' cc bin c t4Ang php thT v F =t bin c t4Ang
php thT _. Gi %T ta bit cc $c %u&t >BiH v >FBiHL i j 1L 2L L n. hi _L $c%u&t c'a bin c F 5c tnh thYA c*ng th+c %aud
n
i i
i 1
>FH >B H.>F B H=
=
)*ng th+c n- 5c gDi c*ng th+c $c %u&t S- '.
V> de &/ )_ 7 h=p b_ng nL t4Ang _ gP ba h=p AKi 1 \i h=p c_ b_ng tt v 1
b_ng $&u. ai h=p AKi 2 \i h=p gP ( b_ng tt v hai b_ng $&u. X&- ngu nhin
=t h=p v t _ 4[t 4a =t b_ng n. FH j >B1H.>FB1H >B2H. >FB2H 0-d 1 2
2>B H a >B H
7 7= =
1 2
1 2 1>F B H >F B H
1l o = = =
/-L ta c_ 1 1 2 2 1 2 1 2
>FH >B H.>F B H >B H.>F B H . .7 1l 7 17l
= + = + = .
V> de $/Q=t * hKt ging 5c ph0n chia AKid XAKi 1 chi 2 % hKt c'a c
*L AKi 2 chi1
4% hKt c'a c *L crn Ki AKi . XAKi 1 c_ t n- S }lL
AKi 2 c_ t n- S ol v AKi c_ t n- S (l. zi t n- S
chung c'a * hKt ging baA nhiu .
Gii6
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1 2
2 1 1>B H >B H >B H
( 12= = =
1 2 >F B H lL}>F B H lLo>F B H lL(= = =
/- 1 1 2 2 >FH >B H.>F B H >B H.>F B H >B H.>F B H= + +
j 2 1 1.lL} .lLo .lL( lL{ ( 1(
+ + =
/ Cng h9 7ay@
?i c`ng gi thit nh5 t4Ang c*ng th+c $c %u&t S- 'L th i6u :in php thT 5c
thZc hin v :t qu bin c F $- 4a. F B H
>B FH>FH
=
B H >F B H >FH7 1l 17l
= = = .
hi _ 1
1. 7 1l>B FH2 2
17l
= = .
V> de L/Q=t - gP AKi inh :ind AKi 1 chi 7L AKi 2 chi 27L AKi
chi (l tng % inh :in c'a tAn b= thit b". mc %u&t hu hzng %au :hAng thMi
gian hAKt =ng nA _ c'a cc AKi t5;ng +ng d 17L 27L 7. Q- ang hAKt =ng
bng b" hzngL tnh $c %u&t @ tng AKi inh :in b" hzng gi thit cc AKi inh :in
:h*ng hzng Png thMiH.
Gii6
GDi F bin c - b" hzngL B i bin c inh :in b" hzng thu=c AKi i 1L H. hi _cc bin c Bi/p nn h S- ' cc bin c. )Sn tnh cc $c %u&t >B iFH.
1
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B H j lL{ >F BH lLo >F BH lL (= = .
7/26/2019 Giao trinh xac suat va thong ke Toan 2013.doc
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$c %u&t @ t4Ang n Sn thZc hin php thT c_ : Sn $u&t hin bin c F 5c tnh thYA
c*ng th+c BY4nAui %au 0-d
: : n :
n n> :H ) p q : lLnq 1 p= = =
V> de &/GiYA ( Sn =t cAn $[c %ec.
a. 2H ) .
o o 21o = =
b.mc %u&t @ t4Ang ( Sn giYA :h*ng c_ Sn nA $u&t hin 3t o ch& : j lH
l (
l
: (
1 7 o27> lH ) .
o o 12o = =
V> de $/
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22
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Ch45ng $/ 7i8n ngBu nhin ; uy EuF de &/ de $/ de /Ben :h*ng hKn ch % Sn vA =t bia n :hi Kn t4[ng ch thR 8ng. GDi
m % vin Kn ben t45t. hi _ m bin ngu nhin.
V> de L/ de L/?i m bin ngu nhin chA y v 89 1. hi _ j m 2 h bin ngu
nhin.
/ Hai i8n ngBu nhin 9 EF 1
o
1
o
1
o
1
o
1
o
1
oV> de $/m 1H >m 2H
) ) 17 ) 17= = = = = = = = =
hi _L bng ph0n phi $c %u&t c'a m
m l 1 2
> 1
}
17
2
17
bH V1 m 2H j >m j lH >m j1H j
1
17
V> de / mc %u&t @ =t ng5Mi thi \ \i :hi thi b]ng i $Y lL. Fnh ta %W thi chA
n :hi \ ?i th*i. GDi m % Sn ng5Mi _ 8Z thi. X/p bng ph0n phi $c %u&t c'a
m.
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$/ H p1 p2 . pn
thR de &/)hA bin ngu nhin m c_ bng ph0n phi $c %u&tm l 1 2
> 1(
1
2
1
(X/p bng ph0n phi $c %u&t c'a m.
Gii6
/ T>nh 9hA6 ph0n phi $c %u&t $H c_ cc tnh ch&t %aud
iH 1H$l 1HGlHG =+=
iiH $H h :h*ng gi
iiiH $H in t9c bn t4iivH HHHm> =1 m 7H< < m lH >m 2H< = < = + =
j 2H lH lL lL j lL{
Cch 2.l m 2H >m 1H >m 2H lL( lL lL{< = = + = = + =
1 m 7H j lL
/H F ?!9 @uA
a/ -nh nghta
/t = $c %u&t c'a bin ngu nhin m in t9c $H 5c $c "nh byi
$H$H$ =
1 nghta6 /t = $c %u&t phn nh +c = t/p t4ung $c %u&t tKi i@ $.
/ T>nh 9hA
Nu m h /t = $c %u&t $H thR
iH
=$
8tHtGH$G
iiHb
a
>a m bH $H8$ < =
Chv J &/
H 6 3t hRnh hDcL $c %u&t chA byi iiH 8in tch hRnh phsng gi?i hKn byi P th"h % - j $HL t49c hAnhL 5Mng thsng $ j aL $ j b.
H ~J 8ng ch+ng inh 5cL i v?i bin ngu nhin in t9c m ta c_ >m j $ AH j l.
~A _d >a m bH >a m bH >a m bH >a m bH < = < = < < =
V> de &/)hA bin ngu nhin m c_ h /t = $c %u&t
2{
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1:hi $ aL bH
$H b al :hi $ aLbH
=
nh 9hA
iH $lH$
iiH 18$H$G =+
Chv J $/Nu =t h $H thza On 2 i6u :in iH v iiH thR n_ h /t = $c
%u&t c'a bin ngu nhin in t9c m.V> de /
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Gii6
a. R h $H in t9c t4i tKi $ j nn ta c_
2
$ $
1i $H j H ia$ H 1 a 1 a
= = =
hi _L ta c_l :hi $ lLH
$H $H 2$ :hi $ lLH
= =
.
b. mc %u&t @ =t :hch hng phi chM :h*ng qu 2 ph[t
(>m 2H > m 2H 2H H
= < = =
hi _L $c %u&t @ t4Ang :hch hng c_ 2 ng5Mi phi chM :h*ng qu 2 ph[t d
2 1
2
( 7 .1o.7> 2H )
{2 = =
lL2
V> de L/ Bin ngu nhin in t9c m c_ h ph0n phi $c %u&t
$
l :hi $ l$H lH
1 Y :hi $ l
= > >
a.
b. mc %u&t cSn tR
Cch 1.~`ng h ph0n phi $c %u&t
> 1 m 1H 1H 1H 1 Y l 1 Y < = = =
Cch 2. ~`ng h /t = $c %u&t
1 l 1
$
1 1 l
1$
l
> 1 m 1H $H8$ l8$ Y 8$
Y 1 Y
< = = +
= =
2
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_/C!9 ha @: f9 \4ng 9a i8n ngBu nhin
&/ K| ;wng [!n
a/ -nh nghta
)hA bin ngu nhin m. C vDng tAn c'a m =t %L :U hiu m v 5c $c "nh
nh5 %aud
w Nu m bin ngu nhin 4Mi 4Kc nh/n cc gi t4" $ iv?i $c %u&t >m j $ iH j pi i
j 1Ln H thR =
=n
1i
iip$m .
w Nu m bin ngu nhin 4Mi 4Kc nh/n cc gi t4" $ iv?i $c %u&t >m j $ iH j pi i
j 1L 2L H thR i ii 1
m $ p
+
== nu chu\i h=i t9H.
w Nu m bin ngu nhin in t9c v?i h /t = $H thR +
= 8$H$G$m nu
tch ph0n h=i t9H.
Chv J &/
H C vDng tAn c'a m gi t4" t4ung bRnh thYA $c %u&t c'a cc gi t4" bin ngu
nhin m nh/n.
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Gii6
nh 9hA 9a W| ;wng [!n
H ) j ) ) h]ng %H
H :mH j :m : h]ng %H
H am bH j am b aL b cc h]ng %H
H Nu mL =c /pd m.H j m.
H )hA =t h nA _ v m bin ngu nhinL ta c_
i i
i mHH $ Hp = nu m bin ngu nhin 4Mi 4Kc
mHH $H $H8$+
= nu m bin ngu nhin in t9c.
V> de L/)hA ph0n phi $c %u&t % - hzng m t4Ang =t ca vicd
m l 1 2
> lL lLl lLl1
aH
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bH Q\i - hzng phi %Ta ht 2 t4iu PngL tnh % ti6n %Ta - t4ung bRnh t4Ang ca
vic.
Gii6
a. I - hzng t4ung bRnh t4Ang =t ca vic
m j l.lLl 1.lLl 2.lLl2 j lL11b. GDi % ti6n %Ta - t4Ang =t ca vicL ta c_d j 2m t4iu PngH.
/- % ti6n %Ta - t4ung bRnh t4Ang =t ca vic
j 2mH j 2.m j lL22 t4iu Png.
V> de R/Q=t c*ng t- bA hi@ bn th bA hi@ v?i nh gi 1ll nghRn Png1
ng5Mi1 n^. Nu ng5Mi tha gia bA hi@ g3p 4'i 4A t4Ang n^ _ thR nh/n 5c %
ti6n bPi th5Mng 1 t4iu Png. Bit 4]ngL thYA thng :L t ng5Mi tha gia bA
hi@ 4'i 4A t4Ang n^ lLl7. m j $ iH j pi i
j 1Ln H thRn
2 2
i i
i 1
m H $ p=
= .
w Nu m bin ngu nhin 4Mi 4Kc nh/n cc gi t4" $ iv?i $c %u&t >m j $ iH j pi i
j 1L 2L H thR 2 2i ii 1
m H $ p+
=
= nu chu\i h=i t9H.
2
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w Nu m bin ngu nhin in t9c v?i h /t = $H thR 2 2m H $ $H8$+
=
nu tch ph0n h=i t9H.
1 nghta6>h5;ng %ai c'a bin ngu nhin 3c t45ng chA = ph0n tn c'a bin ngu nhin $ung
quanh gi t4" m. Nu m ?n ch+ng tz %Z bin =ng c'a m ?n ha- m :h*ng n "nhL
nu m nhz thR %Z bin =ng c'a m nhzL ha- m t5;ng i n "nh.
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m j m2H mH2 j2
21 11lLl72((
(l 1o
/- Ej9h 9hu}n
)^n b/c hai c'a ph5;ng %ai 5c gDi = ch chuEn c'a mL :U hiu mHm = .
!;n v" A = ch chuEn c'a m t4`ng v?i ;n v" c'a m.9/ T>nh 9hA
H )H j l ) h]ng %HH :mH j :2mH : h]ng %HH Nu mL =c /p thRd mH j mH H.
Nh/n $t dw m )H j m ) h]ng %Hw Nu cc bin ngu nhin m1L m2L L mn=c /p v c_ c`ng qu- u/t ph0n
phi $c %u&t v?i m thR ta c_n n
i i2i 1 i 1
1 1 m m m
n n n= =
= =
.V> de ~/)hA m v t5;ng +ng cc bin ngu nhin =c /p @ chf i nhu/n tnh
thYA 7H hng n^ :hi Su t5 vA hai ngnh F v B nA _.Gi %T m j 12L m j 27L j 1(L j o. Q=t ng5Mi Su t5 vA c hai ngnh F
v B thR cSn Za chDn t Su t5 nh5 th nA @ t 4'i 4A nh&t.Gii6GDi a t phSn t4^ vn Su t5 vA ngnh FL :hi _ t phSn t4^ Su t5 vA
ngnh B c'a ng5Mi _ 1V a. GDi i nhu/n c'a ph5;ng n Su t5 n-L ta c_ d j am 1V aH
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H Nu m bin ngu nhin in t9c thR QY8m % 8%aA chA
8
$H8$ lL7
=
V> de R/m $ H j .
7
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_L/ 7i8n ngBu nhin hai 9hiu
&/ Kh!i nij i8n ngBu nhin W 9hiu
m $ L - H p$ L - Hi 1L 1Ln= = = = =
Bng ph0n phi $c %u&t c'a mLH $c "nh nh5 %aud
m
-1 -2 - n >mH
$1 p$1L-1H p$1L-2H p$1L-H p$1L-nH p$1H$2 p$2L-1H p$2L-2H p$2L-H p$2L-nH p$2H $i p$iL-1H p$iL-2H p$iL-H p$iL-H p$iH m p$L-1H p$L-2H p$L-H p$L-nH p$H
>H p-1H p-2H p-H p-nH 1
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-1 -2 n> p-1H p-2H p-nH
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,2H j 12.lL2 1L72.lL( 22.lL1j 2L(
,H j ,2H ,H2j lL1(
/ H m 2L H j 2L H j 1 2 1 1 1a4ctan . a4ctan2 2 2 + +
j
1o
/ T>nh 9hA
ph0n phi $c %u&t c'a bin ngu nhin hai chi6u mL H c_ tnh ch&t.
H l $L -H 1
H h*ng gi i v?i tng bin %d1 2 1 2
$ $ $ L -H $ L -H<
1 2 1 2- - $L - H $L - H<
H L -H l$L H l L H l L H 1 = = = + + =
H 1$L H $H+ = d ph0n phi $c %u&t bin c'a m
2 L -H -H+ = d ph0n phi $c %u&t bin c'a
H >m $L -1 -2H j $L -2H $L -1H
>$1m $2L -1 -2H j $2L -2H $2L -1H $1L -2H $1L -1H
}
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V> de $/ de &/ mLH ~ $L -H8$8- =
H-$
$L -H $L -H8$8-
=
H $L -H8$8- 1+ +
=
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V> de $/aH
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Nh5 v/-d
2
1
2 $ :hi $
$Hl :hi $
>-1$iH >-2$iH >-$iH >-n$iH
t4Ang _6i
i
i
>$ L - H>- $ H
>$ H= ( )i 1L 1Ln= =
X5u U 4]ngd
n
i ni 1 i
i
1 i i
>$ L - H>$ H
>- $ H 1ai 1L >$ H >$ H
=
=
= = = =
V> de &/>h0n phi $c %u&t c'a 5;ng thng t4iu PngH v gi?i tnh m c'a c*ng
nh0n =t c*ng t- nh5 %aud
(1
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m
lL7 1 1L7
Nd l lL1 lL lL2
Nad 1 lLlo lL1} lL1o
$ L - H lL1l 1>- $ H
>$ H lLol o= = =
1 22 1
1
>$ L - H lLl 1>- $ H
>$ H lLol 2= = =
1 1
1
>$ L - H lL2l 1>- $ H
>$ H lLol = = =
/- bng ph0n phi $c %u&t c'a 5;ng thng c'a n c*ng nh0n d
m j $1 lL7 1 1L7
> 1o
1
2
1
x T\4{ng h< Ein e9
Gi %T mL H bin ngu nhin hai chi6u in t9c.
/t = $c %u&t c_ i6u :in c'a m :hi j -L :U hiu $-H 5c $c "nh byid
2
$L -H $L -H $ - H
-H$L-H8$
+
= =
.
de $/Bin ngu nhin in t9c mL H c_ h /t = $c %u&t Png thMi nh5
(2
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%aud2 2 2
2
2 2 2
1:hi $ - 4
$L-H 4
l :hi $ - 4
+
.
$iL -iH j >$iH. >-i$iH j >-iH. >$i-iH i 1L 1Ln= = H
w Nu mL H bin ngu nhin hai chi6u in t9c thRd
1 2 $L -H $H. - $H -H. $ -H= =
Nh5 v/-L ph0n phi $c %u&t c'a h hai bin ngu nhin b]ng tch gia ph0n phi $c
%u&t bin c'a =t thnh phSn v?i ph0n phi $c %u&t c_ i6u :in c'a thnh phSn crn
Ki.
w Nu hai thnh phSn m v =c /p v?i nhau thR ph0n phi $c %u&t c_ i6u :in
cng b]ng ph0n phi $c %u&t :h*ng i6u :inL [c _ ta c_ :t qud
H m v =c /p >$iL -iH j >$iH. >-iH i 1L 1Ln= = H
H m v =c /p $L -H j 1$H.2-H
o/ C!9 ha @: f9 \4ng 9a hj hai i8n ngBu nhin
ax K| ;wng [!n ;
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!i v?i h hai bin ngu nhinL cc tha % 3c t45ng c; bn c'a n_ t45?c ht cc
:C vDng tAn v ph5;ng %ai c'a cc thnh phSn. )c :C vDng tAnL ph5;ng %ai 5c
$c "nh b]ng cc c*ng th+c %au 0-d
w Nu mL H bin ngu nhin 4Mi 4Kcd
n
i i i i
i 1 i 1 1
m $ >$ H $ >$ L - H= = =
= =
n n
i
1 1 i 1
- >- H - >$ L - H= = =
= =
[ ] ( ) n
2 22
i i i i
i 1 i 1 1
m $ m >$ H $ >$ L - H m= = =
= =
( )n n
2 22
i i
1 1 i 1
- m >- H - >$ L - H = = =
= =
w Nu mL H bin ngu nhin in t9cd
1m $ $H8$ $ $L -H8$8-
+ + +
= =
2 - -H8- - $L -H8$8-
+ + +
= =
[ ]
2 2 2
1m $ m $H8$ $ $L -H8$8- mH
+ + +
= = [ ] 2 2 21 - -H8- - $L -H8$8- H
+ + +
= =
xHij< $ L - H m.= =
!i v?i bin ngu nhin in t9cd
)AvmL H $-$L -H8$8- m.+ +
=
((
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m
lL7 1 1L7 >$H
l lL1
l
lL
l
lL2
l
lLo
1 lLlolLl
lL1}lL1}
lL1olL2(
lL(
>-H lL1o lL(} lLo 1
i i
i
$ - >$ L - H lLl lL1} lL2( lL(7= + + =
mHj l.lLlo 1.lL( j lL(
H j lL7.lL1o 1. lL(} 1L7.lLo j 1L1
de $/Bin ngu nhin hai chi6u mL H c_ h /t = $c %u&t Png thMi nh5 %aud
2 2
2 2
1 $ -:hi 1
o ($L-H $ -
l :hi 1 (
+
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Cov(X,Y) xyf(x,y)dxdy E(X).E(Y)+ +
=
R cc h /t = $c %u&t bin 1$H v 2-H i $+ng qua cc t49c tDa =L 8A _
mH j l v H j lL vR thd
2 2
2 2
2 39 x 4 y3 2
2 39 x 4 y
3 2
1Cov(X,Y ) xyf(x,y)dxdy y xdx dy
6
+ +
= =
de /XOi %u&t hng n^ c'a t4i phiu < v c phiu I c'a =t c*ng t- c_ bng
ph0n phi $c %u&t nh5 %aud
V1l l 1l 2l >IH lL1 lL2 lL( lL 1Nu un Su t5 ti6n vA c t4i phiu v c phiu thR nn Su t5 thYA t baA nhiu
@d
aV XOi %u&t :C vDng thu 5c ?n nh&t.
bV != 4'i 4A v6 Oi %u&t nhz nh&t.
Gii6
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a. GDi p t Su t5 chA t4i phiu 0 p 1 H v gDi m Oi %u&t thu 5c :hi Su
t5 chA c t4i phiu v c phiu thR m j p< 1V pHI.
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_R/ C!9 uy EuF
== : j lL 1L .L n.U hiu m BnL pH.
C!9 ha @: f9 \4ng6m j np m j np1VpH A8m j Ad pH1nqnp A +
Cng h9 >nh ?!9 @uA6 > :1 m :2 H j2
1
:i i n i
n
i :
) p 1 pH
=
V> de &/Q=t nh0n vin =t ng- i chA hng y 1l n;i v?i $c %u&t bn 5c hng y
\i n;i b]ng lL2. GDi m % Sn ng5Mi _ bn 5c hng t4Ang ng-.
aH
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T>nh 9hA
bNu m1L m2L ...L mn cc bin ngu nhin =c /p v c_ c`ng ph0n phi :h*ng =t
v?i tha % p thRn
ii 1
m= tu0n thYA u/t ph0n phi nh" th+c BnL pH.
w Nu m1 Bn1L pHL m2 Bn2L pH v m1L m2=c /p thR m1 m2 Bn1 n2L pH
Phn nh 9hA6Nu m1 1!( ) L m2 2!( ) v m1L m2=c /p thR m1 m2 !( )
Chv J ; ?A< ?c
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a. de/
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ph0n phi $c %u&t c'a md
0 khi x a
x a*(x) khia x
a
1 khi x
= <
>
C!9 ha @: f9 \4ng6
2
X
a ( a) aE(X) #(X) 2 12 2 2
+ = = =
ng deng 9a uy EuF
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ng deng 9a uy EuF
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H Nh/n t49c l$ 5Mng ti c/n
H !Kt cZc Ki tKi $ j
H )_ hai i@ und1 1
2 $ 2 $
+
ph0n b $c %u&t
$Hj 8tY2
1 $
2
Ht2
2
C!9 ha @: f9 \4ngd 2 mm QY8m QA8m m = = = = =
/ )uF 9hu}n hza 9hu}n 9x6 N% &x
-nh nghta6 Bin ngu nhin in t9c xL tu0n thYA u/t chuEn h_a c_ h /t =
ph0n phi $c %u&td 2u 2
Y2
1Hu
= u Phe Ee9 @: LH
ph0n b $c %u&t uHj 8tY2
1 u
2
t2
C!9 ha @: f9 \4ngd xx QY8x QA8x 1a px 1a 1= = = = =
H )a de &/ 1H lL(1 2H lL ({{2J J = =
Gi! \ i hmn 9hu}n6Gi t4" t?i hKn chuEn +c U ngh#a c'a bin ngu nhin
chuEn h_a xL :U hiu u d => Hux> t4a gi t4" u Phe Ee9 @: oH.c_ tnh ch&td = 1uu
V> de $/
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>x1Lo(H j lLl7l7 ulLl7l7j 1Lo(.
Chv J6
Nu bit >x bH j + =
Nu bit >x bH j 1+ = v + =
9/ Cng h9 >nh ?!9 @uA
Nu m HGN 2 thR
= AA
w > m V m H j
l2
,u- tec d > m V H j 2AH j lL{(
,u- tec 2 d > m V 2H j 2A2H j lL7((
V> de L/
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Nu m BnL pH v thza Ond 1 p 1 p
n 0,31 p pn
> de R/mc %u&t @ %n phE %au :hi %n $u&t :h*ng 5c :i@ t4a ch&t 5ng b]ng
lL2.
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>
=
l$:hil
l$:hiY.$
2
n2
1
H$G
2
$1
2
n
2
n
U hiu m HnG2
?i +
=l
t1$ 8tYtH$G $ l c_ tnh ch&td k:H1:H$.$H1$ =+=+ : %
ngu-nH v =
2
1.
-nh nghta Wh!96
Nu m1L m2L L m:=c /p v?i nhau v c`ng ph0n phi chuEn Nl 1H thR =:
1iim
H:G2 .
C!9 ha @: f9 \4ng6m HnG2 n2mnm ==
Gi! \ i hmn6 Hn2 $c "nh byi2(n)!(X )> = bng de &/aH H2 > bH ( )(L1}> Hl2 =
bH = =T>nh 9hA6
TC&6Nu m H1alGN thR 2m H1G2
TC$6)hA m HG2 v HnG2 v mL =c /p thR m HnG2 +
Hj u6w )hA miHn2 i i j1L :H v cc mi=c /p v?i nhau thR
=
:
1i
im Hn2
:
1i
i =
7{
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w Nu m1L m2L L m:=c /p v?i nhau v c`ng ph0n phi chuEn Nl 1H thR
=
:
1i
im H:G2 .
/ Chv J6 hi b/c tZ 8A n t^ng thR ph0n phi :hi bRnh ph5;ng %W $&p $f qu- u/t
chuEn.
Nu bit ( )2(n) 2(n)! > = = .
Nu bit ( )2(n) 2(n)1! < = = .R/ Quy EuF
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bH = =
T>nh 9hA6 )hA x Nl 1H v HnG2 v xL =c /p thRn
xm= = = .
Nu bit ( )(n) (n)1! T < = = .Nu n l v :h*ng chA bit (n) ta ch&p nh/n
(n) + =
o/ Quy EuF
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bH = < = =
V> de $/~`ng bng gi t4" t?i hKn HnLn 21 L tR cc $c %u&t %au
aH ( )(3,)! * ,-9> bH ( ){2L(> HL( = > =
bH ( ) ( )(4,9) (4,9) (4,9) (4,9)0,02 0,02! * f 1 ! * f 1 0,02 0,9< = > = =T>nh 9hA6 )hA xL Sn 5t tu0n thYA ph0n phi :hi bRnh ph5;ng b/c t5;ng +ng n 1L
n2v xL =c /p thR2
1
n
nxm= n1L n2H
Chv J6
Nu bit 1 2 1 2(n ,n ) (n ,n )!(* ) f > = = v 2 1(n ,n )
1
1f
=
Nu bit 1 2 1 2(n ,n ) (n ,n )1!(* ) f < = =
ol
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Ph0n ++6 THNG K TON
Ch45ng / C5 @I EJ huy8 Bu
_ &/ C!9 Wh!i nij 95 n
&/ T`ng h ; Bu
w
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m1H j m2H j j mnH j 2
$/ Ph45ng
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_$/ Th:ng W ; f9 \4ng 9a Bu ngBu nhin
&/ Kh!i nij ; h:ng W
ic tng hp u j m1L m2L L mnH 5c thZc hin 85?i 8Kng =t h nA
_ c'a cc m1L m2LL mn. n- 5c gDi thng :L :U hiu Gd
G j m1L m2L L mnH
?i u c9 th@d j $1L $2L L $nH ta c_ g j $1L $2L L $nH
/- thng : =t h c'a cc bin ngu nhin v bn th0n thng : cng
=t bin ngu nhin tu0n thYA qui u/t ph0n phi $c %u&t nh&t "nh v?i cc tha %
3c t45ng nA _.
)hnh thng : c`ng v?i u/t ph0n phi $c %u&t c'a thng : c; %y @ %u-4=ng cc th*ng tin c'a u chA 8&u hiu nghin c+u c'a tng th@.
V> de &/)hA u ngu nhin :ch th5?c n j d j m1L m2L mH.
Q=t % thng : 3
2 21 2 3i
i 1
X X X 1X S (X X)
3 2 =
+ += = .
V> de $/)hA u ngu nhin :ch th5?c n j d j m1L m2L mH.
Q=t % thng : 1 2 3 1 2 31 1
X 2X X X 2X 3X/ /
4 6
+ + + += = .
$/ ' @: h:ng W hng deng 9a Bu ngBu nhin
ax T\ung ]nh Bu6
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-nh EJ &6nu bin ngu nhin gc c_ :C vDng tAn mH j v ph5;ng %ai mH j
2 thR m H j m H jn
2
nHm%Y = .
V> de &/Gi %T t4n =t tng th@ c_ hai u :ch th5?c n j 7d
1j 1 ( ( 1H
2j 1 7 ( o (H
mc "nh t4ung bRnh 1 2x x c'a cc u t4n
Gii6
11 4 3 4 1
x 2,6
+ + + += = 2
1 4 6 4x 4
+ + + += =
x T`ng ]nh
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n2 2
ii 1
1S (X X)
n 1 ==
hA3c = =:
1i
2
ii
2 Hmmn1n
1I
>h5;ng %ai Iw25c $c "nh byi c*ng th+cdn
2 2
i
i 1
1Iw m H
n ==
hA3c
:2 2
i i
i 1
1Iw n m H
n ==
t4Ang _d t4ung bRnh tng th@.
-nh EJ / QI.1n
nI2
= I2H j 2 Iw2H j 2
!= ch chuEn ud =
=n
1i
2i Hmm
1n
1I v
k2
i ii 1
1S n (X X)
n 1 ==
I 5c
gDi = ch u v gi t4" c9 th@ c'a n_ t4n =t gi t4" c9 th@ c'a u :U %d
n2
ii 1
1 n (x x)
n 1 n 1== =
hA3ck
2
i ii 1
1 n n (x x)
n 1 n 1== =
x T0n @uA Bu6 Gi %T t tng th@L &- u ngu nhin :ch th5?c n c_ m phSn tT
ang 8&u hiu nghin c+u. hi _ tSn %u&t u =t thng :L :U hiu 5c $c
"nh byi bi@u th+c %aud n
m=
de &6!i6u t4a 8Aanh thu h]ng n^ c'a o7 8Aanh nghip nhz y =t v`ng thu
5c bng % iu %aud
~Aanh thu t4iu ~NH 27 l ( 7 o { (l
I 8Aanh nghip t5;ng +ng o 1 } {( 1lo }7 l 1l
~&u hiu nghin c+u t4Ang tng th@ 8Aanh thu m t4n =t n^. !@ tin tnh
tAn ta /p bng %aud
o7
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$i ni ni$i ni$i2
1H 2H H (H27l(7o{(l
o1}{(
1lo}7l1l
17ll
127(271o{1llol111ll12l
{7l11{ll(1}2}77((
12}7l11l1ol(1l{l1721l(}ll
o7 12{ll (((oo
{7L(o7
12{ll$n
n
1$
:
1i
ii ===
=t4iu !N
2}L(H{7L(o7
(((oo
H$$nn
1
% 22
n
1i
2
ii === =
l{2L2%%27L(2}L(.o(
o7%
1n
n% 2 ====
=
Nu cc % iu %ai :hc nhau 4&t t hA3c c_ qu nhi6u gi t4"L th5Mng %T 89ng
7ng de $/
~Aanh thu t4iu ~NH 17(V17} 17} V 1o2 1o2 V 1oo 1oo V 1{l 1{l V 1{( 1{(V1{} 1{}V1}2
I 8Aanh nghip t5;ng +ng 1l 1( 2o 2} 12 } 2
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o{
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_/ Quy EuF
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~A _
2
2
2
1
2
1
2121
nn
HHmmx
+
=
Nl 1H.
w )c thng : 21
2
112
1
IH1n
= v 2
2
2
222
2
IH1n
= tu0n thYA ph0n phi :hi bRnh
ph5;ng b/c tZ 8A t5;ng +ng n1 1H v n2 1H.
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bRnh u chnh tSn %u&t v?in
Hp1pHpH
== p $c %u&t c'a tng
th@H
x Nu n ' ?n p 4&t nhz v np gSn b]ng np1VpH thR %W ph0n phi $&p $f
>Ai%%An v?i tha % p.
9xNu 7n> v Llp
p1
p1
p
n
1
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Ch45ng L/ M9 E4ng 9!9 ha @: 9a i8n ngBu nhin
_&/ Ph45ng de &/)hA bin ngu nhin gc m c_ m j m j 2 . )h+ng inh 4]ng 2Iam
Sn 5t 5?c 5ng :h*ng chch c'a v 2 .
Gii6
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Gii6
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V> de L/KTQD $%%"x.)hA u ngu nhin nm1L m2L mH /p t tng th@ ph0n
phi N 2a H. X/p cc thng : 211 m(
1m
2
1m
(
1G ++= 212 m
2
1m
o
1m
1G ++=
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Bi@u th+c $c %u&t tng qut c'a qu- u/t :h*ng =t
>$j p$1V pH1V$
X/p u ngu nhin :ch th5?c nd j m 1L m2L L mnH v gi t4" j $1L $2L L $nH
c'a n_ /p h hp U v?i \i gi t4" $ i$ij l hA3c $ij 1H
X$1L $2L L $nL pH ji$1
i Hp1$p
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_$/ Ph45ng
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Nhn xt:H de &/
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bH ?i 1LlA = thYA c*ng th+c $c "nh :ch th5?c uL :hi = tin c/- :h*ng i ta
c_
}7HoL1H1Ll
1n 2
2
2
=
/- @ t^ng = chnh $c t lL2 n lL1 thR :ch th5?c u t^ng t 27 n }7x Ph45ng @ai 2 9h4a i8 ; W>9h h49 Bu n %
)hDn thng :I
m
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V> de $/!@ $c "nh t4Dng 5ng c'a cc baA b=t t4Ang :hAL ng5Mi ta Y c0n ngu
nhin 17 baA b=t c'a :hA v tR 5c $ jL}:g %2j lL1((. O- tR :hAng tin c/-
i $+ng c'a t4Dng 5ng t4ung bRnh c'a cc baA b=t t4Ang :hA v?i = tin c/- .
Gi thit t4Dng 5ng c'a cc baA b=t bin ngu nhin c_ ph0n phi chuEn.
Gii6
GDi m t4Dng 5ng b=t _ng baAL thYA gi thit m tu0n thYA u/t chuEnd HGN 2 .
/- t4Dng 5ng t4ung bRnh chnh v bi tAn 5a v6 tR :hAng tin c/- i
$+ng
chA d
1(H 17H
lLll7 lLll71 lL7
17 17
S SP X ! X !
< < + = =
?i u c9 th@ ta c_d $ jL}:g {Ll1((Ll% == {{L2t H17
ll7Ll = L hAng tin c/-
i $+ng chA d {{L217
{Ll}L{{L2
17
{Ll}L +
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GDi m chi6u 8i c'a chi titL thYA gi thit m tu0n thYA u/t chuEnd HGN 2 v?i2 ch5a bit. Bi tAn 5a v6 tR :hAng tin c/- i $+ng chA d
2 2
1H 1H 1 lL7n nS S
P X ! X !n n
< < + = =
~A n j 2ll l nn v?i = tin c/- 7 1H 1H2 lLl27 lLl27 1Lon! ! &
= = .
?i u c9 th@ /p bng
$i ni ni$i ni$i2
7(L}
7(L}1
7(L}2
7(L}7(L}(
7(L}7
7(L}o
7(L}{
o
1(
({(7
17
{
2}L}l
{o{L(
1}lLlo
27{{Ll11(o{L}l
1}1lLl7
}22Ll
}(Ll
1}l1}L2(l
(2l7{Ll7
1{2Loo
1(1{Loo17(L17l
2}1L2(2
(71((L2(
21Ll{7Ll1}nj2ll 1lo{Ll7 ol1}lL7l
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w hAng tin c/- bn phi chA 2 d2
2
2 1H
1
1Hl 1
n
n SP
< < =
x9h4a i8 W| ;wng [!n
Gi %T tng th@ m tu0n thYA u/t chuEn HGN 2 . )hDn thng :
w22
2
nIG= =
2 Hn
w hAng tin c/- hai pha chA 2 dw2 w2
2
2 n H 2nH
2 1 2
nI nI> 1
< < =
w hAng tin c/- bn phi chA 2 dw2
2
2nH
nI> 1
< < + =
w hAng tin c/- bn phi chA 2 dw2
2
2nH
1
nI> l 1
< < =
?i u c9 th@L $c "nh %2v t _ $0- 8Zng 5c :hAng tin c/- t5;ng +ng.
Nu 6 bi -u cSu = ch chuEn thR :hai c^n ph5;ng %aid 2=
V> de &/?i gi thit c'a v 89 9c 1 v = tin c/- 7 hO- 5?c 5ng :hAng tin
c/- chA ph5;ng %ai c'a :ch th5?c cc chi tit.
Gii6!0- bi tAn 5?c 5ng ph5;ng %ai c'a bin ngu nhin ph0n phi chuEn
HGN 2 :hi ch5a bit . hAng tin c/- chA 2 d
2 22
2 1H 2 1H
2 1 2
1H 1H1 lL7
n n
n S n S P
< < = =
?i u c9 th@ ta c_ 2 21H 21HlL{7 lLl27lLlll2o} 2ll 1}L} 2}(L}% n = = = =
hi _ :hAng tin c/- chA 2
lll2o.llll1}}Ll}L1}
lll2o}Ll.1
}L2}(
lll2o}Ll.1 22
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Gii6 !0- bi tAn 5?c 5ng ph5;ng %ai c'a bin ngu nhin ph0n phi chuEn
HGN 2 :hi ch5a bit . hAng tin c/- chA 2 d
2 22
2 1H 2 1H
2 1 2
1H 1H1 lL7
n n
n S n S P
< < = =
X/p bng tnh tAn
$i ni $ini $i2ni21 { 1({ l}{
22 1{ {( }22}
2 2 oo{ 17(1
2( 2{ o(} 17772
27 17 {7 {7
2o 7 1l }l&%% $L& RL"o
?i u c9 th@ ta c_ 2 2H 2HlL{7 lLl271Lo1}1 1ll {Lo 12}L(% n = = = =
hi _ :hAng tin c/- chA 2
2 2.1Lo1}1 .1Lo1}1 1L2({( 2L1}o 1L11o}{1 1.({{{l112}L( {Lo
< < < < < u p 1 1n
=
w hAng tin c/- bn t4i chA pd 1 H
> l p u 1n
+ =
7i [!n6ch th5?c u ti thi@u n bA = tin c/- 1V chA t45?c v = 8i
:hAng tin c/- :h*ng v5t qu AchA t45?c d
2
22
l
uH1
n
V> de &/i@ t4a ngu nhin (ll %n phE 8A =t - %n $u&t th&- c_ 2l ph
phE. ?i = tin c/- 7L
aH O- 5?c 5ng :hAng tin c/- i $+ng chA t ph phE c'a nh -
bH O- 5?c 5ng t ph phE ti aL ti thi@u c'a nh -
cH Nu un = chnh $c y c0u aH :h*ng qu lLl1 cSn i6u t4a u ?i b]ng
baA nhiu.
Gii6
GDi p t ph phE c'a - _. Nh5 v/- p c; c&u c'a t/p hp %n phE 8A- %n $u&t 4a thYA 8&u hiu ph phE. !0- bi tAn 5?c 5ng t c'a tng th@
aH w hAng tin c/- i $+ng chA p
2 2
1 H 1 H1 lL7
P & p &
n n
+ = =
?i n j (ll j 2l(ll j lLl7 ulLl27j 1LoL ta c_
lLl7.lL7 lLl7.lL7lLl7 .1Lo lLl7 .1Lo(ll (ll
lLl2}o lLl{1(
p
p
+
bH w hAng tin c/- bn t4i chA p
1 Hl 1 lL7
P p &
n
+ = =
?i n j (ll j 2l(ll j lLl7 ulLl7j 1Lo(7L ta c_
lLl7.lL7l lLl7 .1Lo(7 l lLlo(ll
p p +
}2
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w hAng tin c/- bn phi chA p
1 H1 1 lL7
P & p
n
= =
?i n j (ll j 2l(ll j lLl7 ulLl7j 1Lo(7L ta c_
lLl7.lL7lLl7 .1Lo(7 1 lLl21 1
(llp p
cH 2 222 2l
1 H lLl7.lL7.1Lo 1}2(L}
lLl1
n &
= =
n j1}27
V> de $/!i6u t4a nhu cSu tiu 8`ng =t AKi hng h_a c'a =t v`ng ng5Mi ta nghin
c+u 1ll gia Rnh v th&- c_ ol gia Rnh c_ nhu cSu v6 AKi hng h_a _. )hA bit =
tin c/- 7.aH Bit v`ng _ c_ 2lll h= gia RnhL hO- 5?c 5ng :hAng tin c/- i $+ng chA
% h= gia Rnh t4Ang v`ng c_ nhu cSu v6 AKi hng h_a _.
bH Bit v`ng _ c_ 1lll h= gia Rnh c_ nhu cSu %T 89ng hng h_a t4nL hO- 5?c
5ng % h= gia Rnh c'a v`ng _.
Gii6
GDi N % h= gia Rnh c'a v`ng _L Q % h= gia Rnh c_ nhu cSu v6 AKi hng h_a
t4n.
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w hAng tin c/- 2 pha chA 2 d
w2 w22
2 H 2 H
2 1 2
1n n
nS nS P
< < =
w hAng tin c/- bn t4i chA 2
w22
2 H
1
l 1n
nSP
< < =
w hAng tin c/- bn phi chA 2
w22
2 H1
n
nSP
< < + =
Iau _ tnh tAn t4n uL @ thu 5c %w2 t4a cc gi t4" t?i hKn v :t u/n
1
< < =
w hAng tin c/- bn t4i chA 2
22
2 1H
1
1Hl 1n
n SP
< < =
w hAng tin c/- bn phi chA 2
22
2 1H
1H1
n
n SP
< < + =
Iau _ tnh tAn t4n uL @ thu 5c %2 t4a cc gi t4" t?i hKn v :t u/n
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. . . .n
=
:
1i
ii n$ =
:
1i
i
2
in$
de6Q+c haA ph ngu-n iu chA =t ;n v" %n phE bin ngu nhin m ph0n
phi chuEn HGN 2 . !@ 5?c 5ng = ph0n tn c'a +c haA ph ngu-n iu ng5Mi
ta c0n thT 27 %n phE thu 5c bng % iu %aud
Q+c haA ph gaH 1L7 2lLl 2lL7
I %n phE 7 1} 2
)hA = tin c/- lL hO- 5?c 5ng = ph0n tn A b]ng HL = ph0n tn ti aL ti
thi@u c'a +c haA ph t4Ang 2 t45Mng hpd
aH !O bit m j 2l
bH )h5a bit m
Gii6
!0- bi tAn $0- 8Zng :hAng tin c/- chA 2 4Pi %u- 4a :hAng tin c/- t4Ang
2 t45Mng hp O bit v ch5a bit m.
aH )hDn thng :d2
2w2 I27
G
== H27G2
X/p bng tnh 5cd %w2j lLl{.
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w hAng tin c/- bn phi chA 2 dw2
2
227H
271 lL
SP
< < + = =
.
?i %w2j lLl{ 227HlL 1oL({ = ta c_
w2
2 2227H
lL1
27lLl7l
% < < + < < +
bH )hDn thng :d2
2
2
2(S2
= = H2(G2
X/p bng tnh 5cd %2j lLlo1{.
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Ch45ng R/ Ki nh gi huy8 h:ng W
_&/ Gi huy8 h:ng W
&/ Kh!i nij
Gi thu-t thng : cc nh 6 pht bi@u v6 8Kng ph0n phi $c %u&t c'a binngu nhinL v6 cc tha % c'a bin ngu nhin hA3c v6 tnh =c /p c'a cc bnn.
Gi thu-t thng :L :U hiu Ad gi thu-t gc
Gi thu-t i c'a n_L :U hiu 1
)3p gi thu-t thng : A 1H
V> de &/Nhu cSu t4ung bRnh v6 =t hng h_a n^ na- caA h;n n^ t45?c. Bit
nhu cSu t4ung bRnh n^ t45?c 1lll ;n v"thng. O- pht bi@u c3p gi thu-t
thng :.
Gii6 GDi m nhu cSu v6 AKi hng h_a _d m = . Bi tAn 5a v6 :i@ "nh
c3p gi thu-tdA
1
d 1lll
d 1lll
=
.
$/ Tiu 9hu}n Wi nh ; in !9
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_$/ Ki nh gi huy8 ; W| ;wng [!n
&/ Ki nh gi hi8 ; W| ;wng [!n 9a i8n ngBu nhin 9z
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Gi t4" quan %td }}{Lo7Ll
27H7l2{L(
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x Ch4a
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w
>
=
211
21A
dc
dc. Qi6n bc bz { } >= uxd
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Bc bz AL ch&p nh/n 1d t4Dng 5ng c'a AKi qu t4Png thYA hai ph5;ng php
:hc nhau
bH
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_/ Ki nh gi hi8 ;
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>
=
12dc
12dc
2
1
2
A
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Chv J 6 Gi t4" t?i hKn c'a ph0n phi i%hY4 t4a bng Phe Ee9 @: sH c_ tnh ch&td
HnLnG
1
HnLnG
1|2
21
1
=
V> de $/)_ hai ging [a c_ n^ng %u&t t4ung bRnh $&p $f nh5 nhau %Ang +c = ph0ntn v6 n^ng %u&t c_ th@ :hc nhau. !@ :i@ t4a ng5Mi ta g3t u tKi 2 v`ng t4Png hai
ging [a _ v thu 5c :t qu %aud
Gi:ng Eva S: i gf Ph45ng @ai
F n1j (1 (1L11% 21 =
B n2j l 72Lo% 22 =?i +c U ngh#a 17L hO- chA :t u/n v6 v&n 6 t4nL bit n^ng %u&t hai ging [a t4n
bin ngu nhin c_ ph0n phi chuEn.
Gii6GDi m1L m2 n^ng %u&t c'a hai ging [a F v BL thR ch[ng c_ ph0n phi chuEn.
Bi tAn 5a v6 :i@ "nh c3p gi thu-td
=2
2
2
11
2
2
2
1A
dc
dc
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_L/ Ki nh gi hi8 ; ha @: < 9a i8n ngBu nhin
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:hch hng thR th&- c_ 27l ng5Mi tiu 8`ng AKi %n phE n_i t4n. ?i +c U ngh#a
7L hO- :t u/n v6 hiu qu c'a chin 8"ch qung cA _.
Gii6GDi p t :hch hng tiu 8`ng AKi %n phE y "a ph5;ng F. !0- bi
tAn :i@ "nh tha % p c'a ph0n phi FpH. )3p :i@ "nh gi thu-t thng : c_
8Kngd Ad p j pA 1d plLo
R n j (ll 1llL tiu chuEn :i@ "nhdHp1p
nHpxG
AA
A
== $&p $f Nl 1H
?i o(7L1ul7Ll l7Ll == v i6n bc bz Ho(7L1 +=
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~A p ch5a bit nn tha- byi21
2211
nn
nn
++
= d
1 2
1 2
5/ U
1 1f(1 f)
n n
= =
+
c_ ph0n
phi $&p $f chuEn Nl 1H nu n1 l n2 l
w 211 ppdc d { }2uxdx >=
w 211 ppdc > d { } >= uxdx
w 211 ppdc < d { } de $6i@ t4a ngu nhin cc %n phE c`ng AKi 8A hai - %n $u&t thu 5c
% iu %aud
Nh !y S: S< 49 Wi \a S: =
?i u c9 th@L ta tR 5cd lLlll
ll2Ll
1lll
2l 21 ====
l2oLl
ll1lll
l2l =
+
+=
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}1L1
ll
1
1lll
1{{Ll.l2oLl
lLll2Llx q% =
+
= x q% .
/- ch5a c_ c; %y @ bc bz A.
)_ th@ cAi t ph phE c'a hai nh - nh5 nhau.
1ll
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_ R/ Suy din h:ng W
Bit th*ng tin c'a tng th %u- An th*ng tin c'a u
&/ Suy din ; Bu ngBu nhin \v `ng h8 9z
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=
+
212u7u
1o
1l(lu
n 111
===
2u(7u1o
1l(lu
n 222
==+=
+
l(7oLll22}Llu2uu2121l22}Ll21=+======
H(7a7Gm H j 1V 7((Lll(7oLl1 ==
bH Bi tAn 5a v6 tnh $c %u&t @ (lm> .
IT 89ng c*ng th+c =
+ ===+21 ====== de $/ m nghip %T 89ng =t AKi ngu-n iu v?i 5ng tKp ch&t bin ngu
nhin ph0n phi chuEn v?i ph5;ng %ai 2l ga 2t4Ang 1 :g ngu-n iu.
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ngu-n iu ?i nh/p v6 ng5Mi ta &- ngu nhin 4a 1o:g.
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w?i == 21 l d
+ =
p(1 p)! 0 f p + 1
n{H
:hAng tin c/- chA m
+ p(1 p)
0 f p +n
V> de /Q=t * hng ' tiu chuEn $u&t :hEu nu t %n phE AKi =t c'a * hng
_ b]ng l.
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T+ )+U THY' KHZO
1. !A u P. mc %u&t v thng :