How to Indicate Possible Outcomes for a Discrete Random Variable

By Deborah J. Rumsey

A discrete random variable X can take on a certain set of possible outcomes, and each of those outcomes has a certain statistical probability of occurring. The notation used for any specific outcome is a lowercase x, and the probability of any specific outcome occurring is denoted p(x), which you pronounce “p of x” or “probability of x.”

It signifies the probability that the random variable X takes on a specific value, which you call “little x“, that is, the probability that X=x. For example, say you roll a die and look at the outcome. The random variable X is the outcome of the die (which takes on possible values of 1, 2, . . . , 6). To denote the probability of getting a 1 on a die, you write p(1) or p(X=1).

Statisticians use an uppercase X when they talk about random variables in their general form; for example, “Let X be the outcome of the roll of a single die.” They use lowercase x to represent specific outcomes of the random variable, like X=x where x can be any of the values 1,2,…,6.

A list or function showing all possible values of a discrete random variable, along with their probabilities, is called a probability distribution, p(x). For example, when you roll a single die, the possible outcomes are 1, 2, 3, 4, 5, and 6, and each has a probability of 1/6 (if the die is fair).

As another example, suppose some renters in an apartment building are dog lovers:

40% own one dog

7% own two dogs

3% own three dogs

50% own zero dogs

For X = the number of dogs owned, the probability distribution for X is shown in the following table:

X p(x)
0 0.50
1 0.40
2 0.07
3 0.03