Page 56 - Fundamentals of Probability and Statistics for Engineers

Random Variables and Probability Distributions 39

n
Euclidian space R n . We note here that an analysis involving random variables
is equivalent to considering a random vector having the n random variables as
its components. The notion of a random vector will be used frequently in what
follows, and we will denote them by bold capital letters X, Y, Z,....

3.2 PROBABILITY DISTRIBUTIONS

The behavior of a random variable is characterized by its probability distribu-
tion, that is, by the way probabilities are distributed over the values it assumes.
A probability distribution function and a probability mass function are two
ways to characterize this distribution for a discrete random variable. They are
equivalent in the sense that the knowledge of either one completely specifies
the random variable. The corresponding functions for a continuous random
variable are the probability distribution function, defined in the same way as in
the case of a discrete random variable, and the probability density function.
The definitions of these functions now follow.

3.2.1 PROBABILITY DISTRIBUTION FUNCTION

Given a random experiment with its associated random variable X and given a
x
real number , let us consider the probability of the event fs : X9s) xg, or,
simply, P9X x). This probability is clearly dependent on the assigned value x.
The function

F X x†ˆ P X x†; 3:1†

is defined as the probability distribution function (PDF), or simply the distribu-
tion function , of X . In Equation (3.1), subscript X identifies the random vari-
able. This subscript is sometimes omitted when there is no risk of confusion.
Let us repeat that F X 9x) is simply P9A), the probability of an event A occurring,
the event being X x. This observation ties what we do here with the devel-
opment of Chapter 2.
The PDF is thus the probability that X will assume a value lying in a subset
of S, the subset being point x and all points lying to the ‘left’ of x . As x
increases, the subset covers more of the real line, and the value of PDF
increases until it reaches 1. The PDF of a random variable thus accumulates
probability as x increases, and the name cumulative distribution function (CDF)
is also used for this function.
In view of the definition and the discussion above, we give below some of the
important properties possessed by a PDF.

TLFeBOOK