Probability mass function

A probability mass function (PMF) assigns to each possible value of a discrete random variable the exact probability that the variable equals that value. All probabilities are non-negative and must sum to exactly one.

ByHoang TruongUpdated July 3, 2026

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In a batch of three items, each independently defective with probability 0.1, the number of defects X follows a binomial PMF. The probability of zero defects is 0.729, of one defect 0.243, of two defects 0.027, and of three defects 0.001. These four values are all non-negative and add up to exactly one, covering every possible outcome for X.

Where it fits

TopicProbability & DistributionsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

p(x) = P(X = x)

Variables

$p (x)$: probability mass at value x (dimensionless)
$X$: discrete random variable (dimensionless)
$x$: specific value in the support of X (dimensionless)

LaTeX

\sum_{x} p(x) = 1

Variables

$p (x)$: probability mass at x (dimensionless)

Validity conditions for a PMF: p(x) ≥ 0 for all x in the support, and the probabilities sum to exactly one.

Check yourself

PracticeCORE

A discrete random variable X has PMF: P(X = 0) = 0.3, P(X = 1) = 0.5, P(X = 2) = 0.2. A student claims P(X = 1.5) = (0.5 + 0.2) / 2 = 0.35, interpolating between adjacent support values. What is the fundamental error in this reasoning?

Select an answer to check your understanding.