Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

728 Answers to Selected Exercises

6.4.1(a) 0. 300 , 0. 225 , 0. 350 , 0 .125.

(b) CI forp 2 :(0. 167 , 0 .283)

6.4.2(a)x,y,

1

n+m

[∑n

i=1(xi−x)

(^2) +∑m
i=1(yi−y)
2 ].
(b)[ nxn++mmy,
∑n
i=1(xi−θ̂^1 )
(^2) +∑m
i=1(yi−̂θ^1 )
2
]
(n+m)−^1.
6.4.3θ̂ 1 =min{Xi},n^1
∑n
i=1(Xi−
θ̂ 1 ).
6.4.4θ̂ 1 =min{Xi},
n/log
[∏
n
i=1Xi/
θ̂n 1
]
.
6.4.5(Y 1 +Yn)/ 2 ,(Yn−Y 1 )/2; no.
6.4.6(a)X+1. 282
√
n− 1
n S;
(b) Φ
(
√ c−X
(n−1)/nS
)
.
6.4.7(a) mle is 0.7263, ˆp=0. 76
ArunofBS:(0. 629 , 0 .828).
Via ˆp:(0. 642 , 0 .878).
6.4.8(a) mle is 64.83,x(45)=64. 6
6.4.9Ifny^11 ≤yn^22 ,thenp̂ 1 =ny^11 and
p̂ 2 =ny 22 ;else,p̂ 1 =̂p 2 =ny^11 ++yn^22.
6.5.1t=− 8 .64,p−value = 0.0001.
6.5.2(81. 30004 , 81 .30156).
6.5.3(b) (− 0. 0249 , 0 .1749).
6.5.6(b)c
Pn
i=1X
i^2
Pm
i=1Yi^2
.
6.5.7F=XY.
6.5.8(b)F=x/y=0.3389, Reject.
6.5.9c
[max{−X 1 ,Xn 1 }]n^1 [max{−Y 1 ,Yn 2 }]n^2
[max{−X 1 ,−Y 1 ,Xn 1 ,Yn 2 }]n^1 +n^2 ,
χ^2 (2).
6.6.8The R functionmixnormal,atsite
listed in the Preface
produced these results:
(first row are initial estimates, sec-
ond row are the estimates after 500
iterations):
μ 1 μ 2 σ 1 σ 2 π
105.00 130.00 15.00 25.00 0.600
98.76 133.96 9.88 21.50 0.704

Chapter 7

7.1.4^13 ,^23.

7.1.5δ 1 (y).

7.1.6b=0,does not exist.

7.1.7does not exist.

7.2.8

∏n

i=1[Xi(1−Xi)].

7.2.9(a) n!θ

−r

(n−r)!e

−^1 θ[Pri=1yi+(n−r)yr].

(b)r−^1 [

∑r

i=1yi+(n−r)yr].

7.3.2 60 y^23 (y 5 −y 3 )/θ^5 ;

0 <y 3 <y 5 <θ;

6 y 5 /5;θ^2 /7;θ^2 / 35.

7.3.3θ^12 e−y^1 /θ, 0 <y 2 <y 1 <∞;

y 1 /2;θ^2 / 2.

7.3.5n−^1

∑n

i=1X

2

i;n

− 1 ∑n

i=1Xi;

(n+1)Yn/n.

7.3.6 6 X.

7.4.2(a)X;(b)X

7.4.3Y/n.

7.4.5Y 1 −^1 n.

7.4.7(a) Yes; (b) yes.

7.4.8(a)E(X)=0.

7.4.9(a) max{−Y 1 , 0. 5 Yn};(b)yes;

(c) yes.

7.5.1Y 1 =

∑n

i=1Xi;Y^1 /^4 n;yes.

7.5.4x/α.

7.5.9x.

7.5.11(b)Y 1 /n;(c)θ;(d)Y 1 /n.

7.6.1X

2

−^1 n.

7.6.2Y^2 /(n^2 +2n).