Computing Expectation by Conditioning

• Basic Idea: Compute the expectation or variance of a (complicated)  random variable by conditioning on another random variable.
• In stochastic process, it is often useful to condition on the first event
• Use of the formula
• $E[X] = E[E[X|Y]]$
• $var[X] = var[E[X|Y]] + E[var[X|Y]]$

Example: