Fundamentals of Probability and Statistics for Engineers

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

(a) What is the mean value of the diameter?

(b) What is the mean value of the cross-sectional area,

4.10 Suppose that a random variable X is distributed (arbitrarily) over the interval

Show that:

(a) mX is bounded by the same limits;

(b)

4.11 Show that, given a random variable X, P(X mX )1if^2 X 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):

Determine:

(a) The average waiting time of an arrival,

(b) The average waiting time for an arrival given that a wait is required,

4.13 For the commuter described in Problem 3.21 (page72), 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 withprobability 2/3
return to his original position after 5 minutes of traveling. Assuming that he is at all

FT(t)

1

p

t

Figure 4. 5 Distribution function, FT (t), of waiting times, for Problem 4.12

114 Fundamentals of Probability and Statistics for Engineers

fX…x†ˆ^6 x…^1 x†; for 0x^1 ;

0 ; elsewhere:



/4)X^2?

aXb:

^2 X

ba)^2

4

.

ˆ ˆ  ˆ

FT…t†ˆ^0 ; fort<^0 ;

p‡… 1 p†… 1 et†



; fort 0 :

EfTg.

EfTjT> 0 g.