Page 157 - Schaum's Outlines - Probability, Random Variables And Random Processes

FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS [CHAP. 4

If XI, . . . , X, are pairwise independent, then

Let

Then by Eq. (4.108), we have

If XI, . . . , Xn are pairwise independent, then (Prob. 3.22)

and Eq. (4.1 11) reduces to

Var z a X = z Var(Xi)
a:
( 1 i)

MOMENT GENERATING FUNCTIONS
4.44. Let the moment of a discrete r.v. X be given by

Find the moment generating function of X.
Find P(X = 0) and P(X = 1).

By Eq. (4.41), the moment generating function of X is

By definition (4.40),

Thus, equating Eqs. (4.113) and (4.114), we obtain