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<math display="block">X_{*} (P) (B) = P \left( X^{-1} (B) \right)</math> for <math>B \in \mathcal{E}.</math>
Any probability distribution is a [[probability measure]] on <math> (E, \mathcal{E}) </math> (in general different from <math>P</math>, unless <math>X</math> happens to be the identity map).{{cn|date=May 2025}}
A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function. One of the most general descriptions, which applies for absolutely continuous and discrete variables, is by means of a probability function <math>P \colon \mathcal{A} \to \Reals</math> whose '''input space''' <math>\mathcal{A}</math> is a [[σ-algebra]], and gives a [[real number]] '''probability''' as its output, particularly, a number in <math>[0,1] \subseteq \Reals</math>.
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