# RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

3.1 DEFINITIONS OF A RANDOM VARIABLE

We have already loosely defined the term random variable in Section 1.2 as a random mechanism whose outcomes are real numbers. We have men­tioned discrete and continuous random variables: the discrete random vari­able takes a countable number of real numbers with preassigned prob­abilities, and the continuous random variable takes a continuum of values in the real line according to the rule determined by a density function. Later in this chapter we shall also mention a random variable that is a mixture of these two types. In general, we can simply state

DEFINITION 3.1.1 A random variable is a variable that takes values accord­ing to a certain probability distribution.

When we speak of a “variable,” we think of all the possible values it can take; when we speak of a “random variable,” we think in addition of the probability distribution according to which it takes all possible values. The customary textbook definition of a random variable is as follows:

DEFINITION 3.1.2 A random variable is a real-valued function defined over a sample space.

Defining a random variable as a function has a certain advantage which becomes apparent at a more advanced stage of probability theory. At our level of study, Definition 3.1.1 is just as good. Note that the idea of a

probability distribution is firmly embedded in Definition 3.1.2 as well, for a sample space always has a probability function associated with it and this determines the probability distribution of a particular random variable. In the next section, we shall illustrate how a probability function defined over the events in a sample space determines the probability distribution of a random variable.