Introduction to Probability

Chapter 9

Conditional expectation

Given that you’ve read the earlier chapters, you already know what conditional expectation is: expectation, but using conditional probabilities. This is an essential concept, for reasons analogous to why we need conditional probability:

• Conditional expectation is a powerful tool for calculating expectations. Using strategies such as conditioning on what we wish we knew and first-step analysis, we can often decompose complicated expectation problems into simpler pieces.
• Conditional expectation is a relevant quantity in its own right, allowing us to predict or estimate unknowns based on whatever evidence is currently available. For example, in statistics we often want to predict a response variable (such as ...