## Glossario

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# ProbabilityConditional Expectation

Momento della lettura: ~10 min

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

Example
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 .

Example
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.

Example
With and as defined above, we have and .

Exercise
Find the conditional expectation of given where the pair has density on .

Solution. We calculate the conditional density as

which means that

So

## 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

Exercise
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 indicator of the event that the particle split at time , then

while

Therefore, . By the tower law, we have

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