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.