ISYE 2027-Suppose L is exponentially distributed with mean

ISYE 2027-Suppose L is exponentially distributed with mean

Subject: Mathematics    / Statistics
Question

Suppose L is exponentially distributed with mean $2000.
(a) What is the p.m.f. or p.d.f., whichever is appropriate, of L?
(b) Compute E[(L ? 1000)+]. Hint: one way to do this is just using LOTUS. Another way involves the memoryless property and the law of total expectation. Let X = (L ? 1000)+. From the law of total expectation (see page 57 of our text), E[X] = E[X | A]P(A) + E[X | Ac]P(Ac). What would be a good choice of the event A to make it easy to figure out each of the conditional expectations? (One of the conditional expectations might be zero.)
(c) Suppose L represents a possible loss that you face, and you wish to buy insurance against this loss. If you buy full coverage, the insurance company assumes all risk (that is, they incur the loss L). The premium charged for this insurance is twice the expected loss. How much does the insurance company charge for full coverage?
(d) To reduce your costs, you might decide to have a deductible d. With a deductible, you are splitting the loss L between you and the insurance company. You are responsible for all losses up to d, and then the insurance company covers any additional losses. That is, you are responsible for L ? d, and the insurance company is responsible for (L ? d)+. Compute E[L ? d]. The easiest way is to take advantage the means that you already know and that (L ? d)+ + L ? d = L. 2

(e) Let us assume that the premium is still twice the expected amount that the insurance company will payout for losses. How much is the premium? Assume that d = $1, 000.
(f) In addition to the premium, you may have to pay some amount of money to cover part of the loss L. What is your total expected cost; that is, the cost of the premium plus the expected cost to cover your portion of the loss L?
(g) If you “self-insure”, that is, you do not buy insurance, then you are responsible for the entire loss L. What is your total expected cost if you self-insure?