▾ Set of symbols
▾ Alphabet
▾ Deduction system
▾ Theory
▾ Zermelo-Fraenkel set theory
▾ Set
▾ Subset
▾ Power set
▾ Hyperpower set sequence
▾ Hyperpower set
▾ Hypersubset
▾ Subset algebra
▾ Subset structure
▾ Measurable space
▾ Measure space
▾ Pushforward measure
▾ Distribution measure
Probability distribution measure
Formulation 2
Let $P = (\Omega, \mathcal{F}_{\Omega}, \mathbb{P})$ be a D1159: Probability space such that
(i) $M = (\Xi, \mathcal{F}_{\Xi})$ is a D1108: Measurable space
(ii) $X : \Omega \to \Xi$ is a D202: Random variable from $P$ to $M$
An D4361: Unsigned basic function $\mu : \mathcal{F}_{\Xi} \to [0, \infty]$ is a probability distribution measure for $X$ with respect to $P$ and $M$ if and only if \begin{equation} \forall \, E \in \mathcal{F}_{\Xi} : \mu(E) = \mathbb{P}( X^{-1}(E) ) \end{equation}
Formulation 3
Let $P = (\Omega, \mathcal{F}_{\Omega}, \mathbb{P})$ be a D1159: Probability space such that
(i) $M = (\Xi, \mathcal{F}_{\Xi})$ is a D1108: Measurable space
(ii) $X : \Omega \to \Xi$ is a D202: Random variable from $P$ to $M$
An D4361: Unsigned basic function $\mu : \mathcal{F}_{\Xi} \to [0, \infty]$ is a probability distribution measure for $X$ with respect to $P$ and $M$ if and only if \begin{equation} \forall \, E \in \mathcal{F}_{\Xi} : \mu(E) = \mathbb{P}( X \in E ) \end{equation}
Also known as
Probability distribution