Conditional probability distribution: Difference between revisions

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Let <math>X : \Omega \to E</math> be a <math>(E, \mathcal{E})</math>-valued random variable. For each <math>B \in \mathcal{E}</math>, define <math display="block">\mu_{X \, | \, \mathcal{G}} (B \, |\, \mathcal{G}) = \mathrm{P} (X^{-1}(B) \, | \, \mathcal{G}).</math>For any <math>\omega \in \Omega</math>, the function <math>\mu_{X \, | \mathcal{G}}(\cdot \, | \mathcal{G}) (\omega) : \mathcal{E} \to \mathbb{R}</math> is called the '''conditional probability distribution''' of <math>X</math> given <math>\mathcal{G}</math>. If it is a probability measure on <math>(E, \mathcal{E})</math>, then it is called [[Regular conditional probability|'''regular''']].

For a real-valued random variable (with respect to the Borel <math>\sigma</math>-field <math>\mathcal{R}^1</math> on <math>\mathbb{R}</math>), every conditional probability distribution is regular.<ref>[[#billingsley95|Billingsley (1995)]], p. 439</ref> In this case,<math>E[X \mid \mathcal{G}] = \int_{-\infty}^\infty x \, \mumu_{X \mid \mathcal{G}}(d x, \cdot)</math> almost surely.

=== Relation to conditional expectation ===