Conditional Expectation 11/9

It is often useful to use conditional probabilities for the computation of expected values, the definition of the conditional expectation follows from the definition of conditional probability:

$$\mbox{ For } \mbox{ such that } P(F)>0,\qquad E(X\vert F)= \sum_{x_i}x_i P(x_i\vert F)$$

This used through the following equivalent of the law of total probability: If F_j,j=1...M is a partition of the state space $\Omega$,

$$E(X)=\sum_{j=1}^M E(X\vert F_j)P(F_j)$$

See the book, page 240, for the proof I gave inclass, and the craps example.