Page 134 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 117

Y
X 2

X 1

Figure 4.7 Frame structure, for Problem 4.27

2
4.29 Let X 1 , X 2 , .. . , X n be independent random variables and let and j be the
j
2
respective variance and third central moment of X j . Let and denote the
corresponding quantities for Y, where Y ˆ X 1 ‡ X 2 ‡ ‡ X n .
2
2
2
2
a) Show that ˆ ‡ ‡ ‡ , and ˆ 1 ‡ 2 ‡ ‡ n .
2
n
1
b) Show that this additive property does not apply to the fourth-order or higher-
order central moments.
4.30 Determine the characteristic function corresponding to each of the PDFs given in
Problem 3.1(a)–3.1(e) (page 67). Use it to generate the first two moments and
compare them with results obtained in Problem 4.1. [Let a ˆ 2 in part (e).]
4.31 We have shown that characteristic function X (t) of random variable X facilitates
the determination of the moments of X. Another function M X (t), defined by
tX
M X t†ˆ Efe g;
and called the moment-generating function of X, can also be used to obtain
moments of X. Derive the relationships between M X (t) and the moments of X.
4.32 Let
Y ˆ a 1 X 1 ‡ a 2 X 2 ‡ ‡ a n X n
where X 1 , X 2 , . .., X n are mutually independent. Show that

a n t†:
Y t†ˆ X 1 a 1 t† X 2 a 2 t† ... X n

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