Definition:Probability Distribution
Definition
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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.
In particular: because the p.d. of $X$ is a p.m.f..
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Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $7.8$
