# nLab probability distribution

### Context

#### Measure and probability theory

measure theory

probability theory

# 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 $\rho$ on a measurable space $X$ such that

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

• 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:

## Examples

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