# How to Indicate Possible Outcomes for a Discrete Random Variable

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 |