Definition:Probability Distribution
Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$.
Then the probability distribution of $X$ is the probability measure of $X$ on $\R$.
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 $X$.
However, both of those terms have wider meanings than the field of probability theory, so beware of this, and use the explicit form if there is any danger of confusion.
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.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): distribution
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distribution