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# 2010-2011秋季A及答案

??1????7??????????2010～2011????????????????A????????

??

??????:????????????????A??????:01014016

????:5

????????????

????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

????????????????????????????????????????

20

10

10

10

10

10

10

10

10

????(20??)??????(????2??)

1.??10??????????2????????.??????????3??,??????????3????????1??????????????

7

15

≈0.4667;??????1??????????????

815

≈0.5333.

2.??????????????????0.4,??X????????3??????????????????????????,??X～b(3,0.4);P{X=1|X>0}=.

3.??X????????????F(x).??Y=2X??????????FY(y)=F(y);Z=F(X)～U(0,1).

4.??X????????????????f(x)=1x

e?(x>0),??E(X)=E(X(X?1))=θ(2θ?1).

5.??X1,X2,X3????????????????N(0,σ2),????a=1a(X????,

1+X2+X3)2????χ2

????;??b=2??,bXX2+X3????F????.

??2????7??

????(10??)??????(????????????????????????????????.??????2??)

1.??A??B????????????,??P(A)P(B)>0,??P(B|A)>0.(??)2.??F(x)=P{X≤x},????P{X≥a}=1?F(a).(??)

3.????????????X,Y????????,????????????????,??min(X,Y)??????????????.(??)

4.????????????????{Xn},??????nlim→∞

D(Xn)=0,????????ε>0,????

nlim→∞

P{|Xn?E(Xn)|>ε}=0.(??)

5.??H0??H1????????????????????????????????????,??????????????????????=P{????H0|H1}.(??)

????(10??)??????(??????????????????????????A,B,C??D.??????2??)

1.??P(A)>0,P(B)>0,????A∩B=?,??(C)

(A)A??B????????

(B)A??B????????

(C)A??B????????

(D)A??B??2.??X～π(λ),??P{X(X?1)=0}=(B)

(A)e?λ

(B)(1+λ)e?λ

(C)e?λ

2

(D)λe?λ

2

3.??X～b(m,p),Y～b(n,p),??????????????.??X+Y～(D).

(A)b(mn,2p)

(B)b(mn,p)

(C)b(m+n,2p)(D)b(m+n,p)

4.??????????X～N(1,9),??

EX

??(C)(A)

1(B)

1(C)3

(D)9

5.??X1,X2,···,Xn????????N(μ,σ2)????????????,????μ????,σ>0????.S2????????????????????????.????????????????????????????(B).

(A)??n(Xi?μ)2(B)(n?

1)S2

=1

i(C)

?μ√?μ√(D)

??3????7??????(10??)??????????????,????????????????????????1%??????????0.6,??2%????????????0.4.????????????????????????????10??????.1.(5??)????10????????????????????;

2.(5??)????????10??????????????????????,??????????????????1%,2%???????????????????

??1:??Bk??????????k%(k=1,2),A??“??????10??????????????”.

P(A)=P(B1)P(A|B1)+P(B2)P(A|B2)

(2??)=0.6×0.9910+0.4×0.9810(2??)=0.869458

(1??)

??2:

P(B(Bk)P(A|Bk)

k|A)=

P(A)(2??)P(B0.6×0.9910

1|A)=0.869458

=0.6241

(2??)

P(B2|A)=1?0.6241=0.3759

(1??)

????(10??)??????????X??Y???????,??X????????????

?f(x)=

?kx2,0<x<2

??0,????1.(5??)??????k;

2.(5??)????????????A={X>c}??B={Y>c}????????,??P(A∪B)=34

,??????c.??1:

??

∞??

2

f(x)dx=k

x2dx

(1??)?∞

=83k??=1(2??+1??)

k=

38

(1??)

??4????7??

??2:

P(A∪B)=P(A)+P(B)?P(AB)=P(A)(2?P(A))=34

(1??+1??)

P(A)=

1

2

(1??)

??p=P{X≤c}=c3

,(1??)??c=22

.

(1??)

????(10??)??X??Y????????,??????????????????:

H

H

HYXHH?1

1

H?111

10

3AB

1.(5??)??A,B;

2.(5??)????Z=X+Y????????.

??1:????

?P{X=?1}=

1+

11+

1=

?????P{Y=0}=A+1???(2??)

P{Y=1}=B+

1??24

??

????

????

1??1??

1

4A+12=???1????1211

(1??)

4

B+24=24

??A=14,B=1

8

.

(1??+1??)

??2:????Z????????????1??,????????????1??:

Z?2?1

1

pi

1

1171

??5????7??

????(10??)??????????????????????X??Y,????????????????,??????σ21??σ2

2.

??????????????????????μ.??????μ,????X??????????n1??????,????????;??Y??????????n2??????,????????1.(5??)????????????????a,μ?a=+(1?a)????μ????????????;2.(5??)??a????μ?a??????.??1:

E(?μa)=aE??+(1?a)??

(1??)=aE??+(1?a)E??(2??)=aμ+(1?a)μ=μ

(2??)

??2:??d(a)=D(μa),??

d(a)=a

2σ21

2σ22

n+(1?a)(2??)1n222d??

(a)=2aσ12(1?a)σ2n?

(2??)

1n2

d”(a)=2σ2

12σ22n+>01n2

??d??

(a)=0,??a=n1σ22n(1??)

1σ2+n2σ.1

????(10??)??????X????????????

f(x;θ)=1

2

e?|x?θ|,?∞<x<+∞

????θ>0??????????.??(X1,X2,···,Xn)??????????????????,1.(5??)??θ??????????;2.(3??)????θ??????????????;

3.(2??)??n=3??,??θ????????????????.

??6????7??

??1:

E(X)=

1

??

x?θ|2

xe?|dx(2??)?∞

u==x?θ1??∞2(u+θ)e?|u|du=θ

(2??)

?∞

θ=E(X)θ

?M=(1??)

??2:

L(θ??n)=

f(xi;θ)

(1??)

i=1

1

??

??

=

??n2n

exp?|xi?θ|

(1??)

i=1

l(θ)=?

??n|x?

i?θ|

??i=1

??

??n(1??)

??l(θ)=?nln2?

|xi?θ|???i=1

?

??3:??????l(θ),??????????

??n|xi?θ|.????n=3,(X(1),X(2),X(3))??????????

i=1

??.??d1=X(2)?X?(1),d2=X(3)?X(2).????

???

???2d1+d2+3(X(1)????????3?θ),θ≤X(1)???|x?d?θ),X??1+d2+(X(2)(1)<θ≤X(2)

???i?θ|=

??(1??)

i=1??????d1+d2+(θ?X(2)),X(2)<θ≤X?(3)???????d?1+2d2+3(θ?X(3)),θ>X(3)

???

θ

?L=X(2)(1??)

??:????????????????,??1??.

??7????7??????(10??)??????????????????????????????????????,??????????????????????X～N(μ,σ2).????????????????16??????,????????????????xˉ=502.92,????

2??????????s=12.(??:χ20.975(15)=6.26,χ0.025(15)=27.49,t0.05(15)=1.75)

1.(5??)??σ??????????95%??????????;

2.(5??)??α=0.05??????,??????????????????????????μ>500?

??1:σ??1?α????????????:??????(2??)

(2??)(n?(n?,χ2χ2α/2(n?1)1?α/2(n?1)????????2215×1215×12=,27.496.26????√√78.574,345.048=

=[8.8642,18.5755]1)S21)S2(1??)

??:??????????????,??????????σ2??????????,????1??.

??2:????????????????:

H0:μ≤500,H1:μ>500

????t??????,????????????????????α=0.05,??????????????

W={t|t>tα(n?1)=t0.05(15)=1.75}

??

xˉ?μ0√t=s502.92?500√==0.9733<1.7512

????????H0,??????????μ????????????500.(1??)(1??)(1??)(1??)(1??)