Page 136 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 4) FUNCTIONS OF RANDOM VARIABLES, EXPECTATION, LIMIT THEOREMS 129

Notice the important difference between Eqs. (4.58) 6nd (,4.59). Equation (4.58) tells us how a
sequence of probabilities converges, and Eq. (4.59) tells us how the sequence of r.v,'s behaves in the
limit. The strong law of large numbers tells us that the sequence (xn) is converging to the contant p.

C. The Central Limit Theorem:
The central limit theorem is one of the most remarkable results in probability theory. There are
many versions of this theorem. In its simplest form, the central limit theorem is stated as follows:
Let XI, . . . , X, be a sequence of independent, identically distributed r.v.'s each with mean p and
variance a'. Let

where 1, is defined by Eq. (4.57). Then the distribution of 2, tends to the standard normal as n -, oo;
that is,
lim 2, = N(0; 1)
n-t m

lim F,,(Z) = lim P(Z, I @(z)
z)
=
fl-' aJ fl* W
where @(z) is the cdf of a standard normal r.v. [Eq. (2.54)]. Thus, the central limit theorem tells us
that for large n, the distribution of the sum Sn = XI + . . + Xn is approximately normal regardless of
the form of the distribution of the individual X,'s. Notice how much stronger this theorem is than the
laws of large numbers. In practice, whenever an observed r.v. is known to be a sum of a large number
of r.v.3, then the central limit theorem gives us some justification for assuming that this sum is
normally distributed.

Solved Problems

FUNCTIONS OF ONE RANDOM VARIABLE
4.1. If X is N(p; a2), then show that Z = (X - p)/a is a standard normal r.v.; that is, N(0; 1).

The cdf of Z is

By the change of variable y = (x - p)/a (that is, x = ay + p), we obtain