Page 129 - Schaum's Outlines - Probability, Random Variables And Random Processes

Chapter 4

Functions of Random Variables, Expectation,

Limit Theorems

4.1 INTRODUCTION
In this chapter we study a few basic concepts of functions of random variables and investigate the
expected value of a certain function of a random variable. The techniques of moment generating
functions and characteristic functions, which are very useful in some applications, are presented.
Finally, the laws of large numbers and the central limit theorem, which is one of the most remarkable
results in probability theory, are discussed.

4.2 FUNCTIONS OF ONE RANDOM VARIABLE

A. Random Variable g(X) :
Given a r.v. X and a function g(x), the expression

defines a new r.v. Y. With y a given number, we denote Dy the subset of Rx (range of X) such that
g(x) s y. Then

where (X E Dy) is the event consisting of all outcomes [ such that the point X([) E Dy . Hence

If X is a continuous r.v. with pdf f,(x), then

B. Determination of fYQ) from fx(x):
Let X be a continuous r.v. with pdf f,(x). If the transformation y = g(x) is one-to-one and has the
inverse transformation
x = g-l(y) = NY)
then the pdf of Y is given by (Prob. 4.2)

Note that if g(x) is a continuous monotonic increasing or decreasing function, then the transfor-
mation y = g(x) is one-to-one. If the transformation y = g(x) is not one-to-one, fb) is obtained as
follows: Denoting the real roots of y = g(x) by x,, that is,