Given an event A, define the conditional expectation of Y given A as
where IA is the indicator random variable.
(a) Let Y ∼ Exp(λ). Find E[Y |Y > 1].
(b) An insurance company has a $250 deductible on a claim. Suppose C is the amount of damages claimed by a customer. Let X be the amount that the insurance company will pay on the claim. Suppose C has an exponential distribution with mean 300. That is,
Find E[X], the expected payout by the insurance company.