Conditional probability distribution: Difference between revisions

Revision as of 13:46, 24 March 2020

Revision as of 13:49, 24 March 2020

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

Then the '''[[conditional probability]] given <math>\scriptstyle \mathcal B</math>''' is a function <math>\scriptstyle \operatorname{P}(\cdot\mid\mathcal{B}):\mathcal{A} \times \Omega \to ([0,1)]</math> such that <math>\scriptstyle \operatorname{P}(A\mid\mathcal{B})</math> is the [[conditional expectation]] of the indicator function for <math>A</math>:

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