Page 131 - Fundamentals of Probability and Statistics for Engineers

114 Fundamentals of Probability and Statistics for Engineers

4.9 The diameter of an electronic cable, say X, is random, with pdf

6x 1 x†; for 0 x 1;
f x†ˆ
X 0; elsewhere:
(a) What is the mean value of the diameter?
2
(b) What is the mean value of the cross-sectional area, /4)X ?
4.10 Suppose that a random variable X is distributed (arbitrarily) over the interval
a X b:

Show that:
(a) m X is bounded by the same limits;
b a) 2
2
(b) .
X
4
4.11 Show that, given a random variable X, P(X ˆ m X ) ˆ 1 if 2 X ˆ 0.
4.12 The waiting time T of a customer at an airline ticket counter can be characterized
by a mixed distribution function (see Figure 4.5):
0; for t < 0;

F T t†ˆ t ; for t 0:
p ‡ 1 p† 1 e †
Determine:
(a) The average waiting time of an arrival, EfTg.
(b) The average waiting time for an arrival given that a wait is required,
EfTjT > 0g.
4.13 For the commuter described in Problem 3.21 (page 72), assuming that he or she
makes one of the trains, what is the average arrival time at the destination?
4.14 A trapped miner has to choose one of two directions to find safety. If the miner
goes to the right, then he will return to his original position after 3 minutes. If he
goes to the left, he will with probability 1/3 reach safety and with probability 2/3
return to his original position after 5 minutes of traveling. Assuming that he is at all

F (t)
T

1

p
t

Figure 4.5 Distribution function, F T (t), of waiting times, for Problem 4.12

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