Conditional Expectation

Definition

The conditional expectation of given is

If is a function of and , then

While is a number, is a function of . Before we observe , we do not know the value of , so it is a random variable which we denote .

Theorem (The Rule of Iterated Expectations)

For random variables and , assuming the expectations exists, we have that

More generally, for any function , we have

Definition

The conditional variance is defined as

where .

Theorem

For random variables and ,

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 3.5.