Fundamentals of Probability and Statistics for Engineers

Appendix C: Answers to Selected Problems 383

5.10 a)

fW…w†ˆ

0 : 19

2 a…w=a†^1 =^2

…w=a†^1 =^2

36 : 6

! 7 : 96

exp 

…w=a†^1 =^2

36 : 6

(^2) ! 6 : 96
4
3
(^5) ; forw> 0
0 ; elsewhere
8

<
mWˆ 1 : 71 a 103 ,^2 Wˆ 8 : 05 a^2  105
b) Same as a)
5.12Yis discrete and
pY y†ˆ

Z 1

0

fX…x†dx; foryˆ 1

Z 0

1

fX…x†dx; foryˆ 0

8

>>

><

>>

>:

5.14 a)

fA…a†ˆ

1

0 : 08 r 0 …a†^1 =^2

; for 4… 0 : 99 r 0 †^2 a 4 … 1 : 01 r 0 †^2

0 ; elsewhere

8

<

:

b)

fV…v†ˆ

1

0 : 08 r 0

3 v

4 

 2 = 3

; for

4

3

… 0 : 99 r 0 †^3 v

4

3

… 1 : 01 r 0 †^3

0 ; elsewhere

8

><

>:

5.16 a)

fY…y†ˆ

2 ‡y

4

; for 2 <y 0

2 y

4

; for 0<y 2

0 ; elsewhere

8

>>

>>

<

>>

>>

:

b) Same as a)
5.21

fT…t†ˆ

…a 1 ‡a 2 ‡‡an†e…a^1 ‡a^2 ‡‡an†t; fort> 0

0 ; elsewhere



5.23fYy)ˆ

Z 1

1

fX 2 x 2 )[fX 1 x 2 ‡y)‡fX 1 x 2 y)]dx 2 ,1<y< 1

5.25

fY…y†ˆ

yey

(^2) = 2
; fory 0
0 ; elsewhere