浙江工业大学概率统计09-10_1_试卷答案

1

.W K£z 2©§¡20©¤

1.#A!B ü ¯ §P(A)=0.5§P(B)=0.6§ P(B|A)=0.8,uP()=)µ

P(AB)=P(A)P(B|A)=0.4,

P(AB)=P(B) P(AB)=0.6 0.4=0.2

2. «" '\ódÕá'n#ó ¤§z#ó '¢¬Çþ p§u " ²n#ó \ó  ¬'VÇ

"

)µ²n#ó \ó  ¬=Qz#ó þ ¬§VÇ (1 p)n"

8

1 ex,x≥0

3.#X'©Ù¼ê F(x)=§uX'VÇ Ý¼ê

:0,x<0

¶Y=e2X'êÆÏ" f(x)=)µ

f(x)=F(x)=EY

ex,x≥0

:0,x<01

1

=E(e2X)=e2xexdx=

30

8

"

4.,m¯¥§3I2åʶ ö& ÅGSü 3 9m|D,uz |DÑk ö =k |DTk I å'VÇ

"

)µd YVF§5¶ öSü 3 9m|D' ª¡k35«¶z |DÑk ö =k |DTk I å= |D I

11

å§ |D 2I§ |D 1å§ù$ ª¡k3!×C3×C2=36

«§VÇ

36

=

4"5.#ü ÅgþX Y pÕá ©yÑlëê λ1,λ2'Ñt©Ù,uX+YÑlëê

'Ñt©Ù"

2

)µd Poisson©Ù'S §X+Y～P(λ1+λ2)"y²Xeµ

P(X+Y=n)=

===

Xnk=0

Xnk=0Xnk=0Xnk=0

P(X=k,Y=n k)P(X=k)P(Y=n k)eeλ1

λk1ek!

λ2

kλn 2

(n k)!

k

(λ1+λ2)

1n!n

λk1λ2

n!k!(n k)!

n

kkn Cnλ1λ2

k

=e=e(λ1+λ2)

1X

n!

k=0

(λ1+λ2)(λ1

+λ2)nn!

6.#oNX～N(µ,σ2)§Ù¥σ2 §S2 $) , ½ 8Ý1 α§uoNþ µ' 8«m Ýúª L= ½ ,~¢1 α§Lò

(g ,g¢)

2Stα(n 1)§¨§¨$)Nþ

)µ § oNþ ' 8Y² 1 α' 8«m Stα(n 1),+Stα(n 1)§l «m Ý 8Ý1 α~¢ §°Ýòy\§=«m Ýòg¢"

7.#oNX～N(µ,σ2)§X1,X2,···,X2n´Ù$)§ÚOþU=

1n

i=1(X2i

X2i 2

u1)§

1)

EU=)µÏ E(X2i X2i

=0§

2

1)

E(X2i X2i 1X2n

n

=D(X2i X2i

1)

1)

=D(X2i)+D(X2i

EU

=

=2σ2

21)

E(X2i X2i

=σ2

i=1

8.#8,abÀÅ'¦^cêXÑl ê©Ù§ÙVÇ Ý f(x)=0.1e0.1x,x≥0

§,<ï daÎbÀŧ|^ ê©Ù'

:0,,x<0ÃPÁSµP(X>t+s|x>s)=P(X>t)§d bÀÅ U¦^5c

3

±þ'VÇ

)µdÃPÁS§ùÚ 5bÀÅU¦^5c±þ'VÇ Ó§

1

P(X>5)=0.1e0.1xdx=e0.1 5≈0.61

5

.ÀtK£zK2©§¡20©¤

1.eP(A)=0.75,P(B)=0.25§u¯ A,B m'9X (A) pÕá

B)

pØ N

C)A B

D)ØU(½

)µA),B),C)Ñ´ UuA'§ Y´D)ØU(½"

2.`¯ü<ÕáGéÓ 8s  g§¦ '·¥Ç©y´0.6Ú0.5§y® 8s&Â¥§u8sd` <Â¥'VÇ (A)

0.6

B)

0.375

C)

6/11

D)

)"0.75

)"

)µAv«8s&`Â¥§Bv«8s&¯Â¥§Cv«8s&Â¥§Dv«8sd` <Â¥§u

P(C)=P(A∪B)=1 P(P()=1 0.5×0.5=0.8P(D)=P(=P(A)P(=0.6×0.5=0.3P(D|C)=P(D)(C)=0.375

22

3.# ÅgþX～N(µ1,σ1)§Y～N(µ2,σ2)§ P(|X µ1|<1)>

P(|Y µ2|<1)§uk9Xª(A))µ

P(|X µ1|<1)=P(|σ1<σ2

B)

σ1>σ2

)"C)

µ1<µ2

D)

µ1>µ2

X µ1

|<σ1X µ2

P(|X µ2|<1)=P(||<

σ211

=2Φ() 1σ1σ111

=2Φ() 1σ2σ2

dΦ'üxS§l σ1<σ2"

4.# ÅgþX,Y pÕ᧠ÑÑlsO ©Ù§u(A)C)

P(X+Y>0)=

1

1)"

11B)D)

P(X Y>0)=

P(max(X,Y)>0)=P(min(X,Y)>0)=

)µ· kX+Y～N(0,2)§X Y～N(0,2)§

4

kP(X>0)=P(Y>0)=1§2

P(X+Y>0)=P(X Y>0)=

1

212

P(max(X,Y)>0)≥P(X>0)=

12

P(min(X,Y)>0)=P(X>0,Y>0)=P(X=0)P(Y=0)=5.é?¿ü ÅgþX,Y§e ·K1d'´(

)"

1

4

(a)cov(X,Y)=0;

(b)D(X+Y)=DX+DY;(c)E(XY)=EX·EY;(d)X,Y pÕá"A)

a,b,c

B)

b,c,d

C)

a,c,d

D)

a,b,c,d

)µÕáS´r'(اکÙk9§ cn =Ú©Ù'A k9"

D(X+Y)=DX+DY+2cov(X,Y)cov(X,Y)=E(XY) EX·EY

l a,b,c´1d'"

6.# ÅgþX1,X2,···,Xn(n>1) pÕá þÑl0-1©Ù§V

n

Ǽê p(x)=px(1 p)1 x,x=0,1§ui=1XiÑl'©Ù´(C)

)"

Ñt©ÙP(np)

D)ØU(½

A) ©ÙN(np,np(1 p))B) ©ÙB(n,p)

)µÕáÑl Ó0-1©Ù'ÚÑl ©Ù§ ©ÙÚÑt©Ù´ÙìC©Ù"

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