4. Conditional Probability and Conditional Expectation. Let Be F be such that 0

4. Conditional Probability and Conditional Expectation. Let B E be such that O < P [Bl < I. For a given A recall that the conditional probability of A given B, written P [A 1B], is given by the formula p[AlB] In this exercise we will relate conditional probability with conditional expectation. To do so define the random variables IB(w) Recall that by construction, E IP IBl and E IP [A1. Show that E [IA11n] (w) p[AlB] In(w) +P [Alw] IB&). Thus, the conditional expectation of IA given In is the conditional probability of A given B on B, and the conditional probability of A given Be on W. Hint: What is 00B)?