Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

Answers to Selected Exercises 723

1.8.8$7.80.

1.8.9(a) 2; (b) pdf isy^23 , 1 <y<∞;

(c) 2.

1.8.10^73.

1.8.12(a)^12 ;(c)^12.

1.8.13P[G=−p 0 ]=^13 ,P[G=1−
p 0 ]=^2312 ,
...,P[G=50−p 0 ]=^2312 (0.0045).

1.8.14Range ofG:{ 2 −p 0 , 5 −p 0 , 8 −
p 0 },Probs: 103 , 106 , 101.

1.9.1(a) 1. 5 , 0 .75; (b) 0. 5 , 0 .05;

(c) 2,does not exist.

1.9.2 e

t

2 −et,t <log2;2;2.

1.9.1210; 0; 2;− 30.

1.9.14(a)−^2

√

2

5 ;(b)0;(c)

2

√

2

5.

1.9.16 21 p;^32 ;^52 ;5;50.

1.9.18^3112 ;^167144.

1.9.19E(Xr)=(r+2)! 2.

1.9.20odd moments are 0,E(X^2 n)=

(2n)!.

1.9.24^58 ; 19237.

1.9.27(1−βt)−^1 ,β,β^2.

1.10.3 0. 84.

1.10.4P(|X|≥5) = 0.0067.

Chapter 2

2.1.1^1564 ;0;^12 ;^12.

2.1.2^14.

2.1.7ze−z, 0 <z<∞.

2.1.8−logz, 0 <z< 1.

2.1.9

( 13

x

)( 13

y

)( 26

13 −x−y

)

/

( 52

13

)

,

xandynonnegative integers

such thatx+y≤ 13.

2.1.11^152 x^21 (1−x^21 ), 0 <x 1 <1;

5 x^42 , 0 <x 2 < 1.

2.1.14^23 ;^12 ;^23 ;^12 ;^49 ;yes;^113.

2.1.15 e

t 1 +t 2

(2−et^1 )(2−et^2 ),ti<log 2.

2.1.16(1−t 2 )−^1 (1−t 1 −t 2 )−^2 ,t 2 < 1 ,

t 1 +t 2 <1; no.

2.2.2

12346 9

1

36

4

36

6

36

4

36

12

36

9

36.

2.2.3e−y^1 −y^2 , 0 <yi<∞.

2.2.4 8 y 1 y 23 , 0 <yi< 1.

2.2.6(a)y 1 e−y^1 , 0 <y 1 <∞;

(b) (1−t 1 )−^2 ,t 1 < 1.

2.3.1 63 xx^11 +2+3;^6 x

(^21) +6x 1 +1
2(6x 1 +3)^2.
2.3.2(a) 2,5;
(b) 10x 1 x^22 , 0 <x 1 <x 2 <1;
(c)^1225 ;(d) 1536449.
2.3.3(a)^3 x 42 ;^3 x
(^22)
80 ;
(b) pdf is 7(4/3)^7 y^6 , 0 <y<^34 ;
(c)E(X)=E(Y)=^2132 ;
Var(X 1 )= 15360553 >Var(Y)= 10247.
2.3.8x+1, 0 <x<∞.
2.3.9(a)
( 13
x 1
)( 13
x 2
)( 26
5 −x 1 −x 2
)
/
( 52
5
)
,x 1 ,x 2
nonnegative integers,x 1 +x 2 ≤5;
(c)
( 13
x 2
)( 26
5 −x 1 −x 2
)
/
( 39
5 −x 1
)
,
x 2 ≤ 5 −x 1.
2.3.11(a)x^11 , 0 <x 2 <x 1 <1;
(b) 1−log 2.
2.3.12(b)e−^1.
2.5.1(a) 1; (b)−1; (c) 0.
2.5.2(a)√^7804.
2.5.8 1 , 2 , 1 , 2 , 1.