Conditional probability distribution: Difference between revisions

Revision as of 17:01, 8 January 2021

Revision as of 08:41, 16 January 2021

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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>

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:<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>

A conditional probability is [[Regular conditional probability|'''regular''']] if <math>\scriptstyle \operatorname{P}(\cdot\mid\mathcal{B})(\omega) </math> is also a [[probability measure]] for all ''&omega;''&nbsp;∈&nbsp;''&Omega;''. An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.

* For the trivial sigma algebra <math>\mathcal B= \{\emptyset,\Omega\}</math> the conditional probability is a constant function, <math>\operatorname{P}\!\left( A\mid \{\emptyset,\Omega\} \right) \equiv\operatorname{P}(A).</math>