Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 117YX 2X 14.30 Determine the characteristic function corresponding to each of the PDFs given inProblem 3.1(a)–3.1(e) (page 67). Use it to generate the first two moments andcompare 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 facilitatesthe determination of the moments of X. Another function M X (t), defined by4.32 LetFigure 4.7 Frame structure, for Problem 4.274.29 Let X 1 , X 2 , ..., X n be independent random variables and let 2 j and j be therespective variance and third central moment of X j . Let 2 and denote thecorresponding quantities for Y, where Y ˆ X 1 ‡ X 2 ‡‡X n .a) Show that 2 ˆ 2 1 ‡ 2 2 ‡‡2 n , and ˆ 1 ‡ 2 ‡‡ n .b) Show that this additive property does not apply to the fourth-order or higherordercentral moments.M X …t† ˆEfe tX g;and called the moment-generating function of X, can also be used to obtainmoments of X. Derive the relationships between M X (t) and the moments of X.Y ˆ a 1 X 1 ‡ a 2 X 2 ‡‡a n X nwhere X 1 ,X 2 ,...,X n are mutually independent. Show that Y …t† ˆ X1 …a 1 t† X2 …a 2 t† ... Xn …a n t†:TLFeBOOK