Page 129 - Fundamentals of Probability and Statistics for Engineers

112 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 M X (t) of a random variable X is defined by
tX
M X t†ˆ Efe g:
In comparison with characteristic functions, the use of M X (t) is simpler since it avoids
computations involving complex numbers and it generates moments of X in a similar
fashion. However, there are two disadvantages in using M X (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, Reading, Mas, pp. 210–217.

PROBLEMS

4.1 For each of the probability distribution functions (PDFs) 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 (page 47), 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

8 3
2
> R
< ; for r 0:33;
f r†ˆ 2 2 2
R 0:9996 ‰ ‡ r 1† Š
R
>
0; elsewhere;
: :
where R ˆ 0:0564. Determine the mean of R.
4.5 A target is made of three concentric circles of radii 3 1/2 , 1, and 3 1/2 feet. Shots
within the inner circle count 4 points, within the next ring 3 points, and within
the third ring 2 points. Shots outside of the target count 0. Let R be the

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