Definition:Probability Distribution/Real-Valued Random Variable
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Definition
Let $\tuple {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable on $\tuple {\Omega, \Sigma, \Pr}$.
Then the probability distribution of $X$, written $P_X$, is the pushforward $X_* \Pr$ of $\Pr$, under $X$, on $\tuple {\R, \map \BB \R}$, where $\map \BB \R$ denotes the Borel $\sigma$-algebra on $\R$.
That is:
\(\ds \map {P_X} B\) | \(=\) | \(\ds \map \Pr {X^{-1} \sqbrk B}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \Pr {\set {\omega \in \Omega : \map X \omega \in B} }\) | Definition of Preimage of Mapping | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Pr {X \in B}\) |
for each $B \in \map \BB \R$, where $X^{-1} \sqbrk B$ denotes the pre-image of $B$ under $X$.
Sources
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $6.12$: The 'elementary formula' for expectation
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $7.8$