Page 135 - Schaum's Outlines - Probability, Random Variables And Random Processes

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

The expression of Eq. (4.52) for the continuous case is recognized as the two-dimensional Fourier
transform (with the sign of j reversed) of fxy(x, y). Thus, from the inverse Fourier transform, we have

From Eqs. (4.50) and (4.52), we see that

which are called marginal characteristic functions.
Similarly, we can define the joint characteristic function of n r.v.'s XI, . . . , X,! by

As in the case of the moment generating function, if X,, . . . , X,, are independent, then

C. Lemmas for Characteristic Functions:
As with the moment generating function, we have the following two lemmas:
Lemma 4.3: A distribution function is uniquely determined by its characteristic function.
Lemma 4.4: Given cdfs F(x), F,(x), F,(x), . . . with corresponding characteristic functions Y(o), Y1(o), Y,(o),
. . . , then Fn(x) + F(x) at points of continuity of F(x) if and only if Y,(o) -+ Y(o) for every o.

4.8 THE LAWS OF LARGE NUMBERS AND THE CENTRAL LIMIT THEOREM
A. The Weak Law of Large Numbers:

Let XI, . . . , X, be a sequence of independent, identically distributed r.v.'s each with a finite mean
E(Xi) = p. Let

Then, for any E > 0,

Equation (4.58) is known as the weak law of large numbers, and X, is known as the sample mean.

B. The Strong Law of Large Numbers:
Let XI, . . . , X, be a sequence of independent, identically distributed r.v.'s each with a finite mean
E(Xi) = p. Then, for any E > 0,

where Xn is the sample mean defined by Eq. (4.57). Equation (4.59) is known as the strong law of large
numbers.