Probability and Stochastic Processes

Chapter 5Product Spaces. Conditional Distribution and Conditional Expectation

In this chapter, we look at the following type of problems: If we know something extra about the experiment, how does that change our probability calculations. An important part of statistics (Bayesian statistics) is to build using conditional distributions. However, what about the more complex and abstract notion of conditional expectation? In principle, if we have the conditional distribution, then calculating the conditional expectation is done using the same methodology as in the previous chapter. However, in many cases we are not capable of coming up with a formula for the conditional expectation, yet we may still calculate and work with conditional expectation. This is due to its properties, which we will learn in this chapter.

Why do we need conditional expectation?

Conditional expectation is a fundamental concept in the theory of stochastic processes. The simple idea is the following: suppose we have no information about a certain variable; then our best guess about it would be some sort of regular expectation. However, in real life it often happens that we have some partial information about the random variable (or in time we come to know more about it). Then what we should do is, every time there is new information, the sample space Ω or the -algebra is changing so they need to be recalculated. ...