Page 133 - Schaum's Outlines - Probability, Random Variables And Random Processes

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

D. Conditional Expectation as a Random Variable:
In Sec. 3.8 we defined the conditional expectation of Y given X = x, E(Y 1 x) [Eq. (3.58)], which is,
in general, a function of x, say H(x). Now H(X) is a function of the r.v. X; that is,
H(X) = E(Y 1 X) (4.38)
Thus, E(Y 1 X) is a function of the r.v. X. Note that E(Y I X) has the following property (Prob. 4.38):
I
ECV XI1 = E(Y) (4.39)

4.6 MOMENT GENERATING FUNCTIONS
A. Definition:

The moment generating function of a r.v. X is defined by
(1 etXi~x(xi) (discrete case)
Mx(t) = ~(e") =
ii2yx(x, d~ (continuous case)

where t is a real variable. Note that Mx(t) may not exist for all r.v.'s X. In general, M,(t) will exist
only for those values of t for which the sum or integral of Eq. (4.40) converges absolutely. Suppose
that Mdt) exists. If we express etX formally and take expectation, then

and the kth moment of X is given by
mk = E(Xk) = Mx(k)(0) k = 1, 2, ...

where

B. Joint Moment Generating Function :
The joint moment generating function Mxy(tl, t,) of two r.v.'s X and Y is defined by
Mxr(tl, t2) = E[e(t1X+t2Y) 1 (4.44)

where t1 and t2 are real variables. Proceeding as we did in Eq. (4.41), we can establish that

and the (k, n) joint moment of X and Y is given by

where
In a similar fashion, we can define the joint moment generating function of n r.v.'s XI, . . . , X, by

Mxl ... x,(tl, . . . , t,) = E[e(tlX1+"'+hXn)] (4.48)