Definition:Probability Distribution
Definition
General 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 $X$.
Real-Valued Random Variable
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 $X$.
Also known as
The probability distribution of $X$ may also be called the distribution or law of $X$.
The probability distribution of $X$ may also be denoted $\mu_X$, $\LL_X$ or $\Lambda_X$.
As an abuse of vocabulary, the "probability distribution" of $X$ may refer to its probability mass function or probability density function.
Also see
- Results about probability distributions can be found here.