Fundamentals of Probability and Statistics for Engineers

Further Reading and Comments

As mentioned in Section 4.2, the Chebyshev inequality can be improved upon if some
additional distribution features of a random variable are known beyond its first two
moments. Some generalizations can be found in:

Mallows, C.L., 1956, ‘Generalizations of Tchebycheff’s Inequalities’, J. Royal Statistical
Societies, Series B 18 139–176.
In many introductory texts, the discussion of characteristic functions of random
variables is bypassed in favor of moment-generating functions. The moment-generating
function MX (t) of a random variable X is defined by

In comparison with characteristic functions, the use of MX (t) is simpler since it avoids
co mputations in volving complex numbers and it generates moments of X in a similar
fashion. However, there are two disadvantages in using MX (t). The first is that it
may not exist for all values of t whereas X (t) always exists. In addition, powerful
inversion formulae associated with characteristic functions no longer exist for moment-
generating functions. For a discussion of the moment-generating function, see, for
example:

Meyer, P.L., 1970, Introductory Probability and Statistical Applications, 2nd edn,
Addison-Wesley, R eading, M as, pp. 210–217.

Problems

4.1 For each of the probability distribution functions (PD F s) given in Problem 3.1
(Page 67), determine the mean and variance, if they exist, of its associated random
variable.

4.2 For each of the probability density functions (pdfs) given in Problem 3.4, determine
the mean and variance, if they exist, of its associated random variable.

4.3 According to the PDF given in Example 3.4 (page47), determine the average
duration of a long-distance telephone call.

4.4 It is found that resistance of aircraft structural parts, R, in a nondimensionalized
form, follows the distribution

112 Fundamentals of Probability and Statistics for Engineers

MX…t†ˆEfetXg:

fR…r†ˆ

2 ^3 R

0 : 9996 ‰^2 R‡…r 1 †^2 Š^2

; forr 0 : 33 ;

0 ; elsewhere;

8

><

:>:

whereRˆ 0 :0564. Determinethe mean ofR.

4.5 A targetis madeof three concentriccirclesof radii 31/2, 1, and 31/2feet. Shots
withinthe inner circlecount4 points,withinthe next ring 3 points,and within
the thirdring 2 points. Shots outsideof the targetcount 0. LetR be the