# Definition:Probability Distribution

## Definition

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

Let $X: \Omega \to \R$ be a random variable on $\tuple {\Omega, \Sigma, \Pr}$.

Then the **probability distribution of $X$** is the pushforward $X_* \Pr$ of $\Pr$ on $\tuple {\R, \mathcal B \paren {\R} }$, where $\mathcal B \paren {\R}$ denotes the Borel $\sigma$-algebra on $\R$.

## Also known as

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

It is also known as the probability mass function.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $7.8$