# Definition:Probability Distribution

## Definition

### General Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {S, \Sigma'}$ be a measurable space.

Let $X$ be a random variable on $\tuple {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.

Then the **probability distribution of $X$**, written $P_X$, is the pushforward $X_* \Pr$ of $\Pr$, under $X$, on $\Sigma'$.

That is:

\(\ds \map {P_X} B\) | \(=\) | \(\ds \map \Pr {X^{-1} \sqbrk B}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map \Pr {\set {\omega \in \Omega : \map X \omega \in B} }\) | Definition of Preimage of Mapping | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \Pr {X \in B}\) |

for each $B \in \Sigma'$, where $X^{-1} \sqbrk B$ denotes the pre-image of $B$ under $X$.

### Real-Valued Random Variable

Let $\tuple {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\tuple {\Omega, \Sigma, \Pr}$.

Then the **probability distribution of $X$**, written $P_X$, is the pushforward $X_* \Pr$ of $\Pr$, under $X$, on $\tuple {\R, \map \BB \R}$, where $\map \BB \R$ denotes the Borel $\sigma$-algebra on $\R$.

That is:

\(\ds \map {P_X} B\) | \(=\) | \(\ds \map \Pr {X^{-1} \sqbrk B}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map \Pr {\set {\omega \in \Omega : \map X \omega \in B} }\) | Definition of Preimage of Mapping | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \Pr {X \in B}\) |

for each $B \in \map \BB \R$, where $X^{-1} \sqbrk B$ denotes the pre-image of $B$ under $X$.

## Also known as

The **probability distribution of $X$** may also be called the **distribution** or **law** of $X$.

The probability distribution of $X$ may also be denoted $\mu_X$, $\LL_X$ or $\Lambda_X$.

As an abuse of vocabulary, the "probability distribution" of $X$ may refer to its probability mass function or probability density function.

## Also see

- Results about
**probability distributions**can be found**here**.