pmf

PMF stands for probability mass function. In probability theory and statistics, it describes the distribution of a discrete random variable by specifying the probability that the variable takes each possible value. The PMF fully characterizes the distribution of the variable and is defined on its support, the set of values the variable can assume.

Formally, if X is a discrete random variable with possible values {x1, x2, ...}, the PMF is pX(x) = P(X = x) for x in the support, and pX(x) = 0 for values not in the support. The probabilities are nonnegative and sum to 1: sum over all x of pX(x) = 1. The cumulative distribution function F(x) = P(X ≤ x) can be obtained by F(x) = sum_{t ≤ x} pX(t). The PMF thus determines all probabilities and, by extension, moments of X, such as the expectation E[X] = sum_x x pX(x) and the variance Var(X) = E[X^2] − (E[X])^2.

Common discrete distributions have standard PMFs. For example, a fair six-sided die has pX(k) = 1/6 for k = 1,...,6. A Bernoulli(p) variable has pX(1) = p and pX(0) = 1−p. The Binomial(n, p) distribution and the Poisson(λ) distribution are defined by their respective PMFs and model counts or rare-event occurrences.

PMFs are distinguished from probability density functions, which describe continuous variables. Outside its support, a PMF assigns zero probability to all values.