Editing Probability distribution (section)

== Kolmogorov definition ==
{{Main|Probability space|Probability measure}}

In the [[measure theory|measure-theoretic]] formalization of [[probability theory]], a [[random variable]] is defined as a [[measurable function]] <math>X</math> from a [[probability space]] <math>(\Omega, \mathcal{F}, \mathbb{P})</math> to a [[measurable space]] <math>(\mathcal{X},\mathcal{A})</math>. Given that probabilities of events of the form <math>\{\omega\in\Omega\mid X(\omega)\in A\}</math> satisfy [[Probability axioms|Kolmogorov's probability axioms]], the '''probability distribution of <math>X</math>''' is the [[Pushforward measure|image measure]] <math>X_*\mathbb{P}</math> of <math>X</math> , which is a [[probability measure]] on <math>(\mathcal{X},\mathcal{A})</math> satisfying <math>X_*\mathbb{P} = \mathbb{P}X^{-1}</math>.<ref>{{Cite book|title=Probability theory : an analytic view|last=W.|first=Stroock, Daniel|date=1999|publisher=Cambridge University Press|isbn=978-0521663496|edition= Rev.|location=Cambridge [England]|pages=11|oclc=43953136}}</ref><ref>{{Cite book|title=Foundations of the theory of probability|last=Kolmogorov|first=Andrey|publisher=Chelsea Publishing Company| year=1950|location=New York, USA|pages=21–24|orig-year=1933}}</ref><ref>{{Cite web|url=https://mathcs.clarku.edu/~djoyce/ma217/axioms.pdf|title=Axioms of Probability|last=Joyce|first=David|date=2014|website=Clark University|access-date=December 5, 2019}}</ref>