ABSTRACT
Conditional expectation is a powerful tool for calculating expectations. The projection interpretation is a helpful way to think about many of the properties of conditional expectation. In problems with a recursive structure, the authors can also use first-step analysis for expectations. Conditional expectation is a relevant quantity in its own right, allowing them to predict or estimate unknowns based on whatever evidence is currently available. There are many interesting examples of using wishful thinking to break up an unconditional expectation into conditional expectations. For example, in statistics the authors often want to predict a response variable based on explanatory variables. The law of total probability allows them to get unconditional probabilities by slicing up the sample space and computing conditional probabilities in each slice. The same idea works for computing unconditional expectations.
