nLab
probability distribution

Contents

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.

Often a probability density.

Definition

A probability distribution is a measure ρ\rho on a measurable space XX such that

  • it is positive: UX: Udρ0\forall U \subset X : \int_U d\rho \geq 0;

  • it is normalized: Xdρ=1\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:

Examples

Revised on March 25, 2014 05:18:11 by Urs Schreiber (89.204.155.101)