Probability Mass Function

probability-theory

Definition

Probability Mass Function

Let be a discrete random variable on a probability space , where is finite or countable. Its probability mass function is

It assigns probability mass to each possible value of and satisfies

1. Non-negativity: for every .
2. Normalisation: .

For every , the probability that takes a value in is

Why Event Probabilities Are Sums

The events for distinct values are pairwise disjoint. For any ,

Countable additivity of therefore gives

Taking yields the normalisation condition .

Contrast with a Probability Measure

Warning

A PMF and a probability measure have different inputs:

The PMF assigns mass to an individual value . The distribution assigns probability to a set of values .

Contrast with a Probability Density Function

A PMF value is itself the probability and cannot exceed . For a probability density function, probabilities are obtained by integration, and the density at one point is not itself a probability.

Example

Fair six-sided die

Let be the result of a fair die roll. Its PMF is

The probability of an even result is obtained by summing the relevant masses: