---
tag: 0QB913WM5E
kind: definition
name: Probability distribution
msc: ["60E05"]
defines: 0DVDCMA2DK
uses: [0VGMEBVW1M, 0BVHZXWKCD]
sources:
  - url: https://en.wikipedia.org/wiki/Probability_distribution
    locator: General probability definition
---

## Definition

Let $E$ be a set with a σ-algebra $\mathcal{E}$.
A probability distribution on $(E, \mathcal{E})$ is a [[probability measure|0VGMEBVW1M]] $\mu$ on $(E, \mathcal{E})$.
A probability distribution on $\mathbb{R}$ is a probability measure on $\mathbb{R}$ with its [[Borel σ-algebra|0BVHZXWKCD]] $\mathcal{B}(\mathbb{R})$.

## Remarks

Each random variable $X$ with values in $E$ gives a distribution, its [[law|0TST4MHA3A]] $B \mapsto P(X \in B)$. Conversely, each distribution $\mu$ is the law of some random variable, for example the identity map on $(E, \mathcal{E}, \mu)$.
A distribution on $\mathbb{R}$ is determined by its [[cumulative distribution function|0MXMS3AE13]].
Important types are discrete distributions, given by a [[probability mass function|0WF4EJP91W]], and absolutely continuous distributions, given by a [[probability density function|04YG690N5V]].

## Explanations

### L1 human

A distribution tells you which values are possible and how likely each range of values is.
For a fair die, the distribution gives each of the values $1, \dots, 6$ the probability $1/6$.