Health insurance coverage

Could someone check this very complicated problem?

I believe I have the correct answer for part a, the graph is omitted, part c the Variance and Standard Deviation seem a little high, and finally I'm not sure at all on part d, because the expected value would go to infinity. I especially need help with part d. Thanks!

Let X be the total medical expenses (in 1000's of dollars) incurred by a particular individual during a given year. Although X is a discrete random variable, suppose its distribution is quite well approximated by a continuous distribution with pdf .

a) What is the value of k?

= This integral must equal 1 so k=12/5.

b) Graph the pdf of X.

c) What are the expected value and standard deviation of total medical expenses?

E(X)= Let then and . We now have

d) This individual is covered by an insurance plan that entails a $500 deductible provision (so the first $500 worth of expenses are paid by the individual). Then the plan will pay 80% of any additional expenses exceeding $500, and the maximum payment by the individual (including the deductible amount) is $2500. Let Y denote the amount of this individual's medical expenses paid by the insurance company. What is the expected value of Y?

[Hint: First figure out what value of X corresponds to the maximum out-of-pocket expense of $2500. Then write an expression for Y as a function of X (which involves several different pieces) and calculate the expected value of this function.]

; ; .