DISCRETE RANDOM VARIABLES AND THEIR PROPABILITY DISTRIBUTION
A random variable is a rule or function that assigns exactly one real number to every outcome of
the same sample space. A random variable can either be discrete or continuous. A discrete random
variable assumes a finite or countable number of values. Some examples of discrete random variables
are the number of “heads” in a toss of 2 coins, number of car owners per town, and number of voters per
municipality. If the random variable is not discrete, then it is said to be continuous.
The probability that a discrete random variable will assume a certain value x will depend on the
probability of the events associated with the value x. In particular, the probability that the random
variable X can assume the value x is obtained by the sum of the probabilities of all outcomes that are
assigned to the value x. The collection of all possible values of the discrete random variable together with
its corresponding probability defines a discrete probability distribution. There are several ways to present
rete probability distribution: (1) table; (2) graph; and (3) formula or function.
A discrete random variable (or a discrete probability distribution) is described by presenting its
qualitative characteristics. Two of these are the mean and the variance. The mean or the expected value
of the discrete random variable X, which is also the mean of its probability distribution is denoted by E(X)
or 𝜇𝑥 , and defined as 𝜇𝑥 = 𝐸(𝑋) = 𝑃(𝑋 = 𝑥𝑖 ). On the other hand, the variance of the random variable
and its probability distribution is denoted by Var (X), V(X), or 𝜎𝑥2 and is defined as 𝜎𝑥2 =
𝐸[𝑋 − 𝐸(𝑋)]2 𝑜𝑟 𝜎𝑥2 = 𝐸(𝑋)2 − [𝐸(𝑋)]2 where 𝐸(𝑋)2 = ∑ 𝑥 2 𝑃(𝑋 = 𝑥).
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