nLab
probability distribution

Idea

A probability distribution is a measure used in probability theory whose integral over some subspace of a measurable space is regarded as assigning a probability for some event to take values in this subset.

Definition

A probability distribution is a measure $\rho$ on a measurable space $X$ such that

• it is positive: $\forall U\subset X:{\int }_{U}d\rho \ge 0$;

• it is normalized: ${\int }_{X}d\rho =1$.

Properties

The collection of all probability distributions on a measurable space carries various metric structures that are studied in information geometry:

• information metric

• Wasserstein metric

Examples

• Gaussian probability distribution

Related concepts

• probability space

• expectation value

• moment

• entropy