概率論課后習(xí)題解答

1（1） （3）{(x,y)|x2y21}（4）{5:0,5:1,5:2,5:3,5:2（1）A； （2）A(BCBCBC)或A(B （3）ABCABCABC；（4）ABBCAC或 （5）ABC （6）ABCABCC C

C1

CCC3（1）500.239；（2） CCC

0.298；（3）

6.6

1098766 6（1）541

6

541

547（1）

（2）7

（3）176 {(x,y)|0xa,0yA{(x,y)|xya(xy),

aPSA

(a/2)2/ x a2/

{(x,y)|0x60,0y A{(x

x2 SA2

6010（1）P(AB)1P(AB)1（2）P(AB)1P(AB)1abP(AB)P(BAB)P(B)P(AB)bP(AB)P(A)P(B)P(AB)1abbc1a11（1）P(AB)P(A(B))P(AAB)P(A)P(A、BABPAB)0,PAB)P（2）P(AB)1P(AB)1P(A)P(B)P(（3）A,B恰好有一個發(fā)生可表示為ABP(ABAB)P(AB)P(P(A)P(AB)P(B)P(AB)P(A)P(B)2P(（4）由(2)知，1PAP(BPAB)PAB)P(AB)P(A)P(B)1P(AB)P(AC)P(BC)P(ABAC)P(A(BC))P(5%100955C50CCP 955CP(A|B)P(AB)P(A)P(AB)

P(P(B

1

1P(A)42,P(B|A)3,P(A)1,P(B|A) P(AB)P(A)P(B|A)23 P(B)P(A)P(B|A)P(A)P(B|A)2314 A2000h還能正常工作，B3000h還能正常工作P(A)0.94P(B)0.87,且ABBP(B|A)P(BA)P(B)0.87P( P( Aii次能調(diào)試好，i=1,2,3，B3PA)1PA|A)3P

|AA)9，B

AAAA

1

1 12P(B)P(A1)P(A1A2)P(A1)P(A1)P(A2|123259 PA)3P(B|A)5P(B|A) P(B)P(A)P(B|A)PA)0.96P(B|A)0P(A|B)P(P(A|B)P(AB)P(BP

0)01

P(A|B)P(P(A|B)P(AB)P(B則P(A0 81

1,P( 444 ,P(A) 44C CP(B|A0)0,P(B|P(B|A)3123,P(B|A) P(B)i

P(

|B)0,P(

16|B)P(A1B)P(A1)P(B|A1) 81

736P(

|B)P(A2B)P(A2)P(B|A2) 23

716P(

|B)P(A3B)P(A3)P(B|A3) 81,P(

|B)

7420.6Aii種工藝得到合格品，i=1,2，B：得到一級品P(A1)0.90.80.80.576P(B|A1)0.8,P(A2)0.70.70.49,P(B|A2)P(A2B)P(A2)P(B|A2)0.490.9p0.70.30.8所以可以看作n次試驗(yàn)，n（1） （2）C2n（3）

1Cn10.94n10.06

nnX23456789P隨量X的分布律X01234Pp(1p)2(1p)3隨量X的分布律X P1 8C F(x)|sinx|F(x)|sinx|不是某個隨量的分布函數(shù)NN5．（1）P(Xk)k

12N（2）

2

4C （3）k

C3

1)1,C3隨量X的分布律X P 7．（1）P(Xk)

k（2）pkpk

kP(Xk)P(Xk

e當(dāng)k時pkpk1;可得，當(dāng)kk0P

k

~E1

P(X1000)110001000

3設(shè)Y：3個這樣的元件使用1000小時后，{X1000}發(fā)生的次數(shù)3則Y~B(3e1

P(Y3)C3e131e104

4

44k查9（1）

ek

ek

44k查（2）P(X8)1P(X8)1k

10.9786XX~

1.X~P(500

)，即X~22P(X3)1P(X2)1

1ee

10.9197n：應(yīng)配備的維修人數(shù)，X：300X~ 要使P(Xn1P(XnnnPXn

n12（1）當(dāng)tTt}是不可 ，F(xiàn)T(t)當(dāng)t0FT(tP(Tt1P(Tt1P(N(t01fT(t)

ttT~E()，根據(jù)指數(shù)分布的無性P(T16|T8)P(T8)1P(T8)

dx 1,C1x21x2（2）當(dāng)x1時FX(x當(dāng)1x1時x1xxF(x) 1xx

1arcsinx x1時FX(x

14（1）f(x)dxCe|x|dx0CexdxCexdx2C1,C

1（2）P(0X1)11exdx1 1(1e10

1etdt1

x FX(x)x1etdt11

x0 15（1）

A

x1ee（2）f(x)1ex

x2

x （3）P(X0)F(0)116．T~E(2

1F(t)1e

tt（1）P(T2)1F(2)（2）P(T10|T9)P(T1)1F(1)e17（1）f(x)dxae(x1)2dx

1,a （2）f(x)

1e(x1)2

( 1212

~N(1,1218（1）2 2 （2）P(4X10)1034323.512 2 （3）P(

2)1P(

2)123232 2 （4）P(

2 2 （5）P(XC)P(XC),P(XC)C3C32

2 2CP(X10.050.12)P(X10.050.12)1F(10.17)F110.1710.059.9310.0512 22(2)229772P(

96)196720.023,9672 96722,P(60X84)84726072211 P(X1200)1120016000.96,400 4001.75,400 22（1）Y1XX0123P1111565（2）YX2X014P1523（1）

(y)P(e

y)P(X

y)

2 2

(y)fX(lny)y（2）y0時FYyy0時FYyPX

2(y)fY(y)

24．

1X4P(1X24fY(y)

fX X,YXX0X,Y111321231 1pp.11953（1）Ayexdydx1AyexdydxA1,A

當(dāng)x0,0y1時F(x,y)xy2veudvdueuxv2y(1ex)y0 當(dāng)x0,y1時 0000F(x,y)x12veudvdueuxv20000

1ex(1ex)y

x0,0y F(x,y)

e

x0,y1

x(x,y)64（1）6G

f(x,y) x6dy6(xx2

0x

(x) f(x,y)dy

yy

6dx

yGx0yGxy yf(y) f(x,y)dx

0x1,0y5（1）

1dy 0x f(x) f(x,y)dy

1dx

0y f(y) f(x,y)dx

XX（2） 6（1） f(x,y)dy 00 f(y) f(x,y)dx 00f(x,y)

fX(x)fY(

f(x,y)dydx1Cx3x4 2 4 （2）fX(x)f(x,fY(y)f(x,yBEDCf(x,yBEDC

fX(xfYy),X與Y8（1）

1111f(x,y)

(x,y) 1

當(dāng)x1

或y0,即(xyA2A當(dāng) x0,0y2xA F(x,y)f(u, 當(dāng)1x0y2x1,22F(x,

v14dudvx2x當(dāng)x0,0y1,即(xyC時F(x,y)

4dudv2vvy0y2

004xyy22F(x,y)(2x2yyf(x,

（2）fY|X(y|x)

fX P(Xm)

p2qP(Yn) p2qn2(nP(j

1,Y2)P(

31（1）

（2）

f(y

dy f(y) f(x,y)dx

f(x,y)

fX(x)fY(12．fZ(z)f(x,x只有當(dāng)0x1,0xzzx1,0z13xdx33zf

0013（1）YX 1101234Z6320123P11241n14（1） )(n1

k nP(Yk)Ckn2

pkqn2 kkkP(Zk)P(XYk)P(Xi)P(YkikCik

piqn1iCkipkiqn2kiCiCkipkqn1n2Ck

pkqn1n2

n1kkkZXY~B(n1n2,fX(x)

0x

f(y)

0yXY相互獨(dú)立，fZ(zfX(xfY(z只有當(dāng)0x1,0zx1時,fX(xfY(zx0x1,z1xf

zdx(z)dx2

0z1zG RG

G{(x,y)|x2y2R(x,y)f(x,y)R0 0FZ(z)

X2Y

zPX2Y2z2

f(x,x2y2z2z z當(dāng)0zR時FZ(z00R2rdrdR當(dāng)z0時FZ(z0;當(dāng)zR時FZ(zf

(z)R00

0z17（1）P(X1,Y1)P(X1,YXYYX01pi11401401201401214p.12121（2）P(X1,Y1)P(X（3）P(Z0)P(P(Z1)P(

0,Y

P(

1,Y1)4Z01234P15194M012P2394N012P4194

(x) f(x,y)dy

xex(y1)dx

y (y) f(x,y)dx

(y

00xex(y1)

y (y|x)

f(x,y)

e xe

yY|

0fX 0

yf(x,

xfX|Y(x|y)

fY(

xfX|Y(x|y)

fX(x),X與Y0x

xyf

fY|

(y|x)10 0 0x1,xyf(x,y)

fX(x)

Y|

(y|x)10 0

dxln(1

0y f(y) f(x,y)dx 1

0 0E(X)10.200.310.320.22E(X)22

10.200.32

10.320.22P(Xk)pqk k

q E(X)kpqk1pqk

k

k

1q

X

,i

則X

Xi 1

9

9P(Xi0)110

P(Xi1)110 9E(Xi)110

i 920E(X)E(Xi)10110

Xi第i顆色子出現(xiàn)的點(diǎn)數(shù)i

nn則XXiE(

)1(126)7,E(X)1(122262) 7 D(Xi)

inE(X)n

E(X

)7n,

D(X)nn

D(X

)35E(X)xf(x)dx1x8(1x)7dx E(X2)x2f(x)dx1x28(1x)7dx D(X)E(X2)E(X)2

E(X)

xf

dx,EX

f(x)是偶函數(shù),E(XE(X2)x21e|x|dxx2exdxx2ex2 0002xex2exdx2ex000D(X)E(X2)E(X)2 (x,y圓f(x,y)R0 0 2

x2y 2R

x2yrdrdx2y

f(x,y)dydxRr2dr2

R

0

R

R

2cos2

E(X)

xf(x)dx22

dx,被積函數(shù)是奇函數(shù),EXE(X2)

x2f(x)dx

2cos2x x

dx

x x 221

1sin2x

2 2

2

D(X)E(

)E(X

10（1） iE(

)3000,

~ 1

600)1P(X

600)

10只燈管都不用更換的概率為e0.210因此，10只燈管至少有一只要更換的概率為1eY：10只燈管中要更換的燈管數(shù)量，則Y~B(10,1e0.2E(Y)10(1e0.2D(Y)10(1e0.2)令Y

X,則Y~N

yE(Xn)nE(Yn)n

yne

2nEXn)nE()

2

2dy

20

0

2

E(Xn)

n為偶數(shù),且n00E(XY)1x2(xy)dydx13x2dx00E(XY)1x2xydydx1x3dx

0 x2yX2Y2x2y

f(x,

x2y2xyex2y2

2

0

rdrd0

2r4sin

222sin2d2

r4

dr

r4

drr

3r2er222 2223r 13er2dr 2 E(X)7300,D(X)P(5200X9400)P(X73002100)PXE(X)1D(X) XX~B(1000,0.5),EX)500DX)P(400X600)

)1

2xydyx

0x

y

0yf

002 1 x

0 07E(X)0x44dx8

E(Y) 2 x

02 1

2x6 x6E(XY)00

dydx0x

dx 3

3330333cov(X,Y)E(XY)E(X)E(Y)2E(X2)2

x2

1dxx

1

5E(Y24 4 4DXEX2EX)211，同理D(Y D(X)D(Y

f(x,y)

1x0,x1 2dy2(x Xf(x) XfYy)

1y0E(X)02xx1dx2x3x20 E(X2)02x2x1dxD(X)E(X2)E(X)2同理E(Y1E(Y21D(Y 0 cov(X,Y)E(XY)E(X)E(Y)XY

D(X)D(Y1

2cov(U,V)cov(Xcov(Y,X)2

D(X)25,D(YD(X)D(Ycov(X,D(X)D(Y

D(XY)D(X)D(Y)2cov(X,Y)2536212D(XY)D(X)D(Y)2cov(X,Y)2536212XX,XX2X2X222 22niiE(Y)1E(nX2)1nii

E(X2)1a

n

n

)1

D(X2

1

E(X2)21

a2

a2

2n2na

an

~N

, 2 設(shè)X：一年里的老人數(shù)目，則X~

167.3X~N 1P(X200)

200170

.12

n個螺絲釘，XX~P(X100)1P(X100)

1000.99n n

1000.99n1.65,nXX~

11 11PX2)1PX2)1k

(200

PX2)1PXX~B(10000,0.8，X~N(8000,1600)（1）P(X8100)1（2）bP(00.1Xb)b2004

bm~B(n,3)4 m~N4

mnPpmnPp 0.001n3 n3

3nXX3nX~N(120P(X10)1X1X22X3~N3X42X5X6~NA

1,B

1

X

2X

22Y1~N(,

),Y2~N(,

),Y1Y2~N(0,2)

2S

~

Z

2(Y1Y2)~2(Y12(Y1Y22S2

~F(m,

1~F(n,m)P(XF(m,n))P(X

(n,m))1P(X

)1

P(X

)

(m,n)

(n,4（1）

X

~N

X P(0.2X0.2)

（2）PX

0.11PX

X n 1 1 1n 1 1 1 220.1n

n

nn

n19.62384.16n38510X10X

~S

10X P(SXS)1

1010X

10

10t0.05(9)

9S

~

P(S

9

(9)

9S

~

9S 9S P(S) 91 99S

99

(9)

2

x3 21．E(X)02

x(x)dx

2

x2

33332．E(X)

xe

dxxe

dxa

b

aba

abxab

E(X2)a

xb

bdxx

a

xe

dxa22bE(Xa22ab2bD(X)E(X2)E(X)2b2n1Sn

nn

n1S3（1）E(X)1xdxx

1（2（2）L() idlnL()n

ln

nnlnL()nln(1)ln n n

lnXi

X~

P(Xk)

k 07 110

212 3iL()P(Xk)e e i

e e 43 5

e e

e42

lnL()4280lnln(212682431202dlnL()4280?80X~G(

E(X)1p

?X ( L(,)nnx lnL(,)nlnnln(1)ln i

dlnL(,)nnln

lnx

dlnL(,)n

lnxinln

2 1 2

)n

E

EXnEX n1 iiEX2ii

)

2

1

1

E D(X)E

nn2n

DXi

EXi

2n2E(S2)

(22)n

即S2是的無偏估EX(1)S2E(X)(1)E(S2)(1)即X1)S2也是的無偏估計(jì)XY~N(1212的極大似然估計(jì)量?X（2）

~N(,n11n1

~N(,4n22n2

X

~N(,n n

4D(?)D(XY)1

1

nnn

nn

可得 n 11令n1nnnn,可使?的方差達(dá)到最小11 E()1E(X)1E(X)1E(

)111E(X)E(X

2626

)111E(X)E(X E()111E(X)E(X2433 24331,2,3都是EX)的無偏估計(jì)量D()111D(X)19D(X4 4

)111D(X)3D(X4 4

)111D(X)1D(X99 993更有效n2n

f(x)dx

fnn nnlim

2f(x)dx1

E

2nnn2 nn

2

1，即

是0.02,n25,X1.02,10.95X UX

nn,X nn查表得2

u0.0