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 $X$.


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