Probability Distribution of Discrete Random Variables

To see how a probability distribution is built, consider throwing two fair dice and noting the numbers on their upper faces. The sample space has $n(S) = 36$ equally likely ordered pairs, $S = \{(1,1), (1,2), \ldots, (6,6)\}$. Let $X$ be the sum of the two numbers shown in a single throw; then $X$ can take any of the 11 values $\{2, 3, \ldots, 12\}$. Each value corresponds to a specific event, for example $[X=2] = \{(1,1)\}$ and $[X=3] = \{(1,2), (2,1)\}$, continuing up to $[X=12] = \{(6,6)\}$. Because all 36 outcomes are equally likely for fair dice, each has probability $1/36$, so $P[X=2] = P\{(1,1)\} = 1/36$, $P[X=3] = P\{(1,2),(2,1)\} = 2/36$, $P[X=4] = 3/36$, and so on. Listing every value of $X$ together with its probability produces Table 7.1 — this pairing of values with probabilities is called the probability distribution of the random variable $X$.

Generalising this idea: the probability distribution of a discrete random variable $X$ is the system of numbers obtained by listing its possible values $x_1, x_2, x_3, \ldots$ alongside the corresponding probabilities $p_1, p_2, p_3, \ldots$, where $p_i = P[X = x_i]$ for $i = 1, 2, 3, \ldots$. A discrete random variable may have finitely many or (countably) infinitely many possible values, but the values are always countable. The distribution is often written as ordered pairs $(x_1, p_1), (x_2, p_2), (x_3, p_3), \ldots$, or, more commonly, laid out in the two-row tabular form of Table 7.2. …