Definition:Probability Distribution

Definition

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Let $X: \Omega \to \R$ be a random variable on $\left({\Omega, \Sigma, \Pr}\right)$.

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

Also known as

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

Sources

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