Page 397 - Fundamentals of Probability and Statistics for Engineers

380 Fundamentals of Probability and Statistics for Engineers

CHAPTER 3

3.1 (a) a ˆ 1,

1; for x ˆ 5
p x†ˆ
0; elsewhere
1
(c) a ˆ 2, p x) , for x ˆ 1, 2, ...
2 x
(e) a > 0,
a 1
ax ; for 0 x 1
f x†ˆ
0; elsewhere
or a ˆ 0,

1; for x ˆ 0
p x†ˆ
0; elsewhere
(g) a ˆ 1/2, neither pdf nor pmf exists
3.2 (a) 1, 1/3, 63/64, 1 e 6a ,1,1, 2 e 1/3 )/2
a
(b) 1, 1, 127/128, e a/2 e 7a ,1 1/2) , 1/2, e 1/4 e 7/2 )/2
3.4 (a)
8
0; for x < 90
<
F X x†ˆ 0:1x 9; for 90 x < 100
1; for x 100
:
(b)
8
0; for x < 0
<
2
F X x†ˆ 2x x ; for 0 x 1
1; for x > 1
:
1 1 1
(c) F X x) ˆ tan x ‡ , for 1 < x < 1
2
3.6 2/3
3.9
0; for x < 0
8
x
>
<
F X x†ˆ ; for 0 x b
b
>
:
1; for x > b
8
1
< ; for 0 x b
f x†ˆ b
X
0; elsewhere
:
3.11 3/a
3.12 b) 1/6

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