Category:Examples of Probability Distributions

This category contains examples of Probability Distribution.

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:

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:

for each $B \in \map \BB \R$, where $X^{-1} \sqbrk B$ denotes the pre-image of $B$ under $X$.