Probability Distribution of a Discrete Random Variable

A discrete random variable is one whose range is either a finite list of numbers or a list that can be written out one after another (countably infinite), such as $0, 1, 2, 3, \dots$. (A random variable that can take any real value in some interval, such as a height or a time, is called continuous instead — that case is not treated here.)

Suppose a discrete random variable $X$ has possible values $x_1, x_2, x_3, \dots$. For each value $x_i$, write $P(X = x_i)$, often shortened to $P(x_i)$, for the probability that $X$ equals $x_i$. These numbers must obey two conditions, exactly like any set of probabilities attached to mutually exclusive, exhaustive events:

$P(x_i) \ge 0$ for every $i$, and $\sum_i P(x_i) = 1$.

The rule that lists every possible value of $X$ alongside its probability is called the probability distribution of $X$. It is usually displayed as a table:

This is the probability analogue of a frequency table: instead of splitting a total count among the different values a variable takes, it splits the total probability of $1$ among them.

Worked Example. Continue the two-dice experiment from the previous section, where $X$ is the sum of the two numbers shown. Since each of the $36$ equally likely outcomes has probability $\frac{1}{36}$, and we just count how many outcomes give each sum, the full distribution of $X$ is: