Discrete random variables

DEFINITION. A random variable X is discrete if its range isa finite or countably infinite set. That is, there exists a setS = {s1, s2, · · ·} such that P (X ∈ S) = 1.

From the definition above, we can deduce that the probabilitydistribution of a discrete random variable is completely determinedby specifying P (X = x) for all x.

DEFINITION. The frequency function of a discrete randomvariable X is defined by

f(x) = P (X = x).

The frequency function of a discrete random variable is known bymany other names: some examples are probability mass function,probability function and density function. We will reserve the term“density function” for continuous random variables.

Given the frequency function f(x), we can determine the distri-bution function:

F (x) = P (X ≤ x) =∑t≤x

f(t).

Thus F (x) is a step function with jumps of height f(x1), f(x2), · · ·occurring at the points x1, x2, · · ·. Likewise, we have

P (X ∈ A) =∑x∈A

f(x);

in the special case where A = (−∞,∞), we obtain

1 = P (−∞ < X <∞) =∑

−∞<x<∞f(x).

EXAMPLE 1.12: Consider an experiment consisting of indepen-dent trials where each trial can result in one of two possible out-comes (for example, success or failure). We will also assume that theprobability of success remains constant from trial to trial; we willdenote this probability by θ where 0 < θ < 1. Such an experimentis sometimes referred to as Bernoulli trials.

We can define several random variables from a sequence of Ber-noulli trials. For example, consider an experiment consisting of n

c© 2000 by Chapman & Hall/CRC