Definition:Probability Distribution/General Definition
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Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\struct {S, \Sigma'}$ be a measurable space.
Let $X$ be a random variable on $\tuple {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.
Then the probability distribution of $X$, written $P_X$, is the pushforward $X_* \Pr$ of $\Pr$, under $X$, on $\Sigma'$.
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 $B \in \Sigma'$, where $X^{-1} \sqbrk B$ denotes the pre-image of $B$ under $X$.
Sources
- 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $10$: Probability: $10.1$: Basics