giao trinh xac suat va thong ke toan 2013.doc

7/26/2019 Giao trinh xac suat va thong ke Toan 2013.doc

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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


3/105

)*+ 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


4/105

Chng ?. 3i@< Anh gi !h&,! !hng D*

GiA t4Rnh Sn Su 4a et bKn Dc nn :h*ng th@ t4nh cc %ai %_t.


5/105

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


17/105

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.


18/105

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


19/105

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


20/105

B H j lL{ >F BH lLo >F BH lL (= = .


21/105

$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 $/


22/105

22


23/105

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.


26/105

$/ 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{


28/105

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 /


29/105

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


30/105

_/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 .


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.


31/105

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


32/105

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=

= .


j 1L 2L H thR 2 2i ii 1

m H $ p+

=

= nu chu\i h=i t9H.

2


33/105

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.


34/105

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


35/105

H Nu m bin ngu nhin in t9c thR QY8m % 8%aA chA

8

$H8$ lL7

=

V> de R/m $ H j .

7


36/105

_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


37/105

-1 -2 n> p-1H p-2H p-nH


38/105

,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

}


39/105

V> de $/ de &/ mLH ~ $L -H8$8- =

H-$

$L -H $L -H8$8-

=

H $L -H8$8- 1+ +

=


40/105

V> de $/aH


41/105

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


42/105

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


43/105

%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 ;


44/105

!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.+ +

=

((


45/105


46/105

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 (

+


47/105

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


48/105

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.


49/105

_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


50/105

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


51/105

a. de/


52/105

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


53/105

ng deng 9a uy EuF


54/105

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 $/


55/105

>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/


56/105

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.


57/105

>

=

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{


58/105

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


59/105

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


60/105

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


61/105

Ph0n ++6 THNG K TON

Ch45ng / C5 @I EJ huy8 Bu

_ &/ C!9 Wh!i nij 95 n

&/ T`ng h ; Bu

w


62/105

m1H j m2H j j mnH j 2

$/ Ph45ng


63/105

_$/ 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


64/105

-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


65/105

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


66/105

$i ni ni$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


67/105

o{


68/105

_/ Quy EuF


69/105

~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.


70/105

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


71/105

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


72/105

Gii6


73/105

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 ++=


74/105

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


75/105

_$/ Ph45ng


76/105

Nhn xt:H de &/


77/105

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


78/105

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 +


79/105

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


80/105

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

< < + =


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


81/105

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


83/105

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.


84/105

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


85/105

. . . .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{.


86/105


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{.


87/105

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


88/105

_$/ Ki nh gi huy8 ; W| ;wng [!n

&/ Ki nh gi hi8 ; W| ;wng [!n 9a i8n ngBu nhin 9z


89/105


90/105

Gi t4" quan %td }}{Lo7Ll

27H7l2{L(


91/105

x Ch4a


92/105

w

>

=

211

21A

dc

dc. Qi6n bc bz { } >= uxd


93/105

Bc bz AL ch&p nh/n 1d t4Dng 5ng c'a AKi qu t4Png thYA hai ph5;ng php

:hc nhau

bH


94/105

_/ Ki nh gi hi8 ;


95/105

>

=

12dc

12dc

2

1

2

A


96/105

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


97/105

_L/ Ki nh gi hi8 ; ha @: < 9a i8n ngBu nhin


98/105

: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 +=


99/105

~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 =

+

+=


100/105

}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


101/105

_ 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


102/105

=

+

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.


103/105

ngu-n iu ?i nh/p v6 ng5Mi ta &- ngu nhin 4a 1o:g.


104/105

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.


105/105

T+ )+U THY' KHZO

1. !A u P. mc %u&t v thng :

giao trinh xac suat va thong ke toan 2013.doc

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