PMFs

PMF stands for probability mass function. In probability theory, a PMF describes the distribution of a discrete random variable X by assigning a probability to each value that X can take. Formally, p(x) = P(X = x) for every x in the support of X. The function is defined on a countable set of values, satisfies p(x) ≥ 0 for all x, and the sum of p(x) over all x in the support equals 1. Values outside the support have p(x) = 0.

The PMF uniquely determines the distribution of X. From it one can derive the cumulative distribution function F(x) = P(X ≤ x) by summing p(y) over all y ≤ x. Moments are computed as E[X] = sum x p(x) over the support, and Var(X) = E[X^2] − (E[X])^2, with sums taken over the support.

Common examples include: a fair six-sided die, where p(1) = … = p(6) = 1/6 and p(x) = 0 otherwise; a Bernoulli variable X ∈ {0,1} with P(X=1) = p; and distributions such as Binomial, Geometric, and Poisson, each with its own PMF.

PMFs apply only to discrete random variables. For continuous variables, the analogous concept is the probability density function (PDF), not the PMF. In general, a PMF must satisfy sum_x p(x) = 1, while a PDF satisfies integral over its domain equals 1. Estimation of PMFs from data often uses frequency counts or likelihood-based approaches.