Probability mass function

Definition

Let $X$ be a discrete random variable on a probability space $(\Omega, \mathcal{F}, P)$. The probability mass function of $X$ is the function $p_X \colon \mathbb{R} \to [0, 1]$ with $$p_X(x) = P(X = x) \quad \text{for } x \in \mathbb{R}.$$ More generally, a probability mass function on a countable set $S$ is a function $p \colon S \to [0, 1]$ with $\sum_{s \in S} p(s) = 1$.

Remarks

The set $\{x : p_X(x) > 0\}$ is countable, and $\sum_x p_X(x) = 1$. For each Borel set $B$, $P(X \in B) = \sum_{x \in B} p_X(x)$. For a fair die, $p_X(k) = 1/6$ for $k = 1, \dots, 6$ and $p_X(x) = 0$ for other $x$.

Explanations

L1 human

The probability mass function is a table: each possible value with its probability. The entries are never negative and they add up to $1$.