Fundamentals of Probability and Statistics for Engineers

4.30 Determine the characteristic function corresponding to each of the PDFs given in
Problem 3.1(a)–3.1(e) (page 67). U se it to generate the first two moments and
compare them with results obtained in Problem 4.1. [Let a 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 MX (t), defined by

and called the moment-generating function of X, can also be used to obtain

moments of X. Derive the relationships between MX (t) and the moments of X.

4.32 Let

where X 1 ,X 2 ,...,Xn are mutually independent. Show that

X 2

X 1

Y

Figure 4.7 Frame structure, for Problem 4.27

Expectations and Moments 117

4.29 LetX 1 ,X 2 ,...,Xnbe independentrandomvariablesand let^2 j andjbe the
respectivevarianceand third centralmomentofXj. Let^2 anddenotethe
correspondingquantitiesforY, whereYˆX 1 ‡X 2 ‡‡Xn.
a) Showthat^2 ˆ^21 ‡^22 ‡‡^2 n, andˆ 1 ‡ 2 ‡‡n.
b) Show that this additivepropertydoes not apply to the fourth-orderor higher-
order centralmoments.

ˆ



MX…t†ˆEfetXg;

Yˆa 1 X 1 ‡a 2 X 2 ‡‡anXn

Y…t†ˆX 1 …a 1 t†X 2 …a 2 t†...Xn…ant†: