Conditional probability distribution: Difference between revisions

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Given a <math>\sigma</math>-field <math>\mathcal{G} \subseteq \mathcal{F}</math>, the conditional probability <math> \operatorname{P}(A\mid\mathcal{G})</math> is a version of the [[conditional expectation]] of the indicator function for <math>A</math>:

:<math>\operatorname{P}(A\mid\mathcal{BG}) = \operatorname{E}(\mathbf{1}_A\mid\mathcal{BG}) \; </math>

An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.