Conditional probability distribution: Difference between revisions

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==Measure-theoretic formulation==

Let <math>(\Omega, \mathcal{F}, P)</math> be a probability space, <math>\mathcal{G} \subseteq \mathcal{F}</math> a <math>\sigma</math>-field in <math>\mathcal{F}</math>, and <math>X : \Omega \to \mathbb{R}</math> a real-valued random variable (measurable with respect to the Borel <math>\sigma</math>-field <math>\mathcal{R}^1</math> on <math>\mathbb{R}</math>). Given <math>A\in \mathcal{F}</math>, the [[Radon-Nikodym theorem]] implies that there is<ref>[[#billingsley95|Billingsley (1995)]], p. 430</ref> a <math>\mathcal{G}</math>-measurable integrable random variable <math>P(A\mid\mathcal{G}):\Omega\to \mathbb{R}</math> so that <math>\int_G P(A\mid\mathcal{G})(\omega) dP(\omega)=P(A\cap G)</math> for every <math>G\in \mathcal{G}</math>, and such a random variable is uniquely defined up to sets of probability zero. It can then be shown that there exists<ref>[[#billingsley95|Billingsley (1995)]], p. 439</ref> a function <math>\mu : \mathcal{R}^1 \times \Omega \to \mathbb{R}</math> such that <math>\mu(\cdot, \omega)</math> is a probability measure on <math>\mathcal{R}^1</math> for each <math>\omega \in \Omega</math> (i.e., it is [[Regular conditional probability|'''regular''']]) and <math>\mu(H, \cdot) = P(X \in^{-1}( H )\mid \mathcal{G})</math> {{Definition needed|date=June 2017}} (almost surely) for every <math>H \in \mathcal{R}^1</math>. For any <math>\omega \in \Omega</math>, the function <math>\mu(\cdot, \omega) : \mathcal{R}^1 \to \mathbb{R}</math> is called a '''[[Conditional expectation#Definition of conditional probability|conditional probability]] distribution''' of <math>X</math> given <math>\mathcal{G}</math>. In this case,

:<math>E[X \mid \mathcal{G}] = \int_{-\infty}^\infty x \, \mu(d x, \cdot)</math>

almost surely.