Page 134 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 41 FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 127

from which the various moments can be computed. If X,, . . . , X, are independent, then

C. Lemmas for Moment Generating Functions:
Two important lemmas concerning moment generating functions are stated in the following:
Lemma 4.1: If two r.v.'s have the same moment generating functions, then they must have the same
distribution.
Lemma 4.2: Given cdfs F(x), Fl(x), F,(x), . . . with corresponding moment generating functions M(t), M,(t),
M,(t), . . . , then Fn(x) -, F(x) if M,(t) -+ M(t).

4.7 CHARACTERISTIC FUNCTIONS

A. Definition:
The characteristic function of a r.v. X is defined by
(x Brnxipx(xi) (discrete case)

[J ejiwx fX(x) dx (continuous case)
-a,
where o is a real variable and j = p. Note that Yx(w) is obtained by replacing t in Mx(t) by jw if
Mx(t) exists. Thus, the characteristic function has all the properties of the moment generating func-
tion. Now

for the discrete case and

for the continuous case. Thus, the characteristic function Yx(u) is always defined even if the moment
function Mx(t) is not (Prob. 4.58). Note that Yx(w) of Eq. (4.50) for the continuous case is the Fourier
transform (with the sign of j reversed) of fx(x). Because of this fact, if Yx(w) is known, fx(x) can be
found from the inverse Fourier transform; that is,

B. Joint Characteristic Functions:

The joint characteristic function Yx,(w,, w,) of two r.v.'s X and Y is defined by

'1 ej(~lxi+~2~k) PXY(~~, ~k) (discrete case)
i k
C" C" ej(a~x +WZY) fXy(x, y) dx dy (continuous case)

where w, and cu, are real variables.