Conditional Expectation

This chapter presents the concept of conditional expectation. This is a very important concept that generalizes the concept of conditional probability, when conditioning occurs not only on a single event, but on an entire family of events. The chapter reviews the definition of conditional probability with respect to an event with strictly positive probability. Conditional probability accounts for the information brought in by the occurrence of one event on the probability of the occurrence of another event. The chapter provides the general definition for conditional expectations and its chief properties. It also provides a geometric interpretation as an orthogonal projection in the particular cases of square integrable random variables. The chapter explores the relations between conditional expectation and independence. Conditional expectation may be interpreted as an orthogonal projection.