Definition:Probability Distribution

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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