The conditional expectation of given is defined to be the expectation of calculated with respect to its conditional distribution given . For example, if and are continuous random variables, then
Suppose that is the function which returns for any point in the triangle with vertices , , and and otherwise returns 0. If has joint pdf , then the conditional density of given is the mean of the uniform distribution on the segment , which is .
The conditional variance of given is defined to be the variance of with respect to its conditional distribution of given .
Continuing with the example above, the conditional density of given is the variance of the uniform distribution on the segment , which is .
We can regard the conditional expectation of given as a random variable, denoted by coming up with a formula for for each , and then substituting for . And likewise for conditional variance.
With and as defined above, we have and .
Find the conditional expectation of given where the pair has density on .
Solution. We calculate the conditional density as
which means that
The tower law
Conditional expectation can be helpful for calculating expectations, because of the tower law.
Theorem (Tower law of conditional expectation)
If and are random variables defined on a probability space, then
Consider a particle which splits into two particles with probability at time . At time , each extant particle splits into two particles independently with probability .
Find the expected number of particles extant just after time . Hint: define to be or depending on whether the particle splits at time , and use the tower law with .
Solution. If is the number of particles and is the
Therefore, . By the tower law, we have