Page 398 - Fundamentals of Probability and Statistics for Engineers

Appendix C: Answers to Selected Problems 381

0:6; for x ˆ 1 0:6; for y ˆ 1

3:13 a): i) p x†ˆ 0:4; for x ˆ 2 p y†ˆ 0:4; for y ˆ 2
X
Y
x y
iii) e ; for x 0 e ; for y 0
f x†ˆ f y†ˆ
X 0; elsewhere Y 0; elsewhere
b): i) No, iii) Yes
3.17 [F X x) F X 100)]/[1 F X 100)], x 100
3.19 a) 0.087, b) 0.3174, c) 0.274
3.22 0.0039
3.26 a)
0:016; for x ˆ 1
8
>
0:035; for x ˆ 2
>
>
>
>
0:080; for x ˆ 3
>
>
<
p x†ˆ 0:125; for x ˆ 4
X 3
0:415; for x ˆ 5
>
>
>
>
>
> 0:192; for x ˆ 6
:
0:137; for x ˆ 7
b) Table of p i, j)
X 4 X 3
i j
1 2 3 4 5 6 7
1 0.006 0.004 0.003 0.003 0.004 0.000 0.000
2 0.002 0.009 0.008 0.005 0.010 0.002 0.001
3 0.003 0.008 0.015 0.014 0.031 0.008 0.005
4 0.001 0.004 0.015 0.027 0.051 0.017 0.011
5 0.002 0.007 0.029 0.054 0.196 0.075 0.050
6 0.001 0.002 0.005 0.015 0.071 0.060 0.032
7 0.000 0.001 0.005 0.008 0.052 0.030 0.038
CHAPTER 4
2
4.1 a) 5, 0; c) 2, 2; e) a/ a ‡ 1), a/[ a ‡ 1) a ‡ 2)]; g) 1, 3
4.32.44 min
4.6 a) 1/2, b) 2, 4; c) 0, 1
4.12 a) 1 p)/ , b) 1/
4.14 24 min
4.16 P jX 1j 0:75) 0:41 by the Chebyshev inequality, P jX 1j 0:75)
ˆ 0:75
4.19 a) P 55 X 85) 0
b) P 55 X 85) 5/9, much more improved bound

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