Conditional probability distribution: Difference between revisions

Revision as of 08:41, 16 January 2021

Revision as of 08:41, 16 January 2021

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

In other words, <math>\scriptstyle \operatorname{P}(A\mid\mathcal{B}) </math> is a <math>\scriptstyle \mathcal B</math>-measurable function satisfying

:<math>\int_B \operatorname{P}(A\mid\mathcal{B}) (\omega) \, \mathrm{d} \operatorname{P}(\omega) = \operatorname{P} (A \cap B) \qquad \text{for all} \quad A \in \mathcal{A}, B \in \mathcal{B}. </math>