Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

Answers to Selected Exercises 727

4.9.11μ 0 ;n−^1

∑n

i=1(xi−x)

(^2).
4.10.1 8.
4.10.4(a) Beta(n−j+1,j);
(b) Beta(n−j+i− 1 ,j−i+2).
4.10.51!3!4!10! v 1 v^32 (1−v 1 −v 2 )^4 ,
0 <v 2 ,v 1 +v 2 < 1.

Chapter 5

5.1.9No;Yn−n^1.

5.2.1Degenerate atμ.

5.2.2Gamma(α=1,β=1).

5.2.3Gamma(α=1,β=1).

5.2.4Gamma(α=2,β=1).

5.2.7Degenerate atβ.

5.2.9 0. 682.
pchisq(60,50)

• pchisq(40,50)=.686

5.2.10Download functioncdistplt4.

5.2.11(a)1-pbinom(55,60,.95)=0.820
(b) 0. 815.

5.2.14Degenerate atμ 2 +σσ 12 (x−μ 1 ).

5.2.15(b)N(0,1).

5.2.17(b)N(0,1).

5.2.20^15.

5.3.2 0. 954.

5.3.3 0. 604.

5.3.4 0. 840.

5.3.5 0. 728.

5.3.7 0. 08.

5.3.9 0. 267.

Chapter 6

6.1.1(a)θˆ=X/ 4 .(c) 5.03

6.1.2(a)−n/log(

∏n

i=1Xi).

(b)Y 1 =min{X 1 ,...,Xn}.

6.1.4(a)Yn=max{X 1 ,...,Xn}.

(b) (2n+1)/(2n).

(c)

√

1 / 2 Yn.

6.1.5(a)X=θU^1 /^2 ,Uis unf(0,1).

(b) 7.7, 5.4.

6.1.6 1 −exp{− 2 /X}.

6.1.7p̂= 12553 ,

∑ 5

x=3

( 5

x

)

p̂x(1−p̂)^5 −x., 0.3597.

6.1.8(b)− 0 .534.

6.1.9x^2 e−x/ 2 ., 0.2699.

6.1.10max

{ 1

2 ,X

}

.

6.2.7(a)θ^42.

(c)

√

n(θˆ−θ)

D

→N(0,θ^2 /4).

(d) 5. 03 ± 0 .99.

6.2.8(a) 21 θ 2.

6.2.13(b)θˆ=3.547.

(c) (2. 39 , 4 .92), Yes.

6.2.14(a)F(x)=1−[θ^3 /(x+θ)^3 ].

(b)g=function(n,t){u=runif(n)

t*((1-u)^(-1/3)-1)}

6.3.1(b) Test-Stat = 17.28, Reject

6.3.2γ(θ)=P[χ^2 (2n)<(θ 0 /θ)c 1 ]

+P[χ^2 (2n)>(θ 0 /θ)c 2 ].

6.3.8Reject if 2

∑n

i=1Yi<χ

2

1 −α/ 2 (2n)

or

2

∑n

i=1Yi>χ

2

α/ 2 (2n).

6.3.16(a)

( 1

3 x

)nx( 2

3(1−x)

)n−nx

.

6.3.17(a)χ^2 W={

√

nI(X)(X−θ 0 )}^2.

(b) Downloadwaldpois.R.

(c)χ^2 W=6.90,p−value = 0.0172.

6.3.18

(

x/α

β 0

)nα

×exp

{

−

∑n

i=1xi

(

1

β 0 −

α

x

)}

.