Question

An insurance agent sells a policy which has a $100 deductible and a $5000 cap. This means that when the policy holder files a claim, the policy holder must pay the first $100. After the first $100, the insurance company pays the rest of the claim up to a maximum payment of $5000. Any excess must be paid by the policy holder. Suppose that the dollar amount X of a claim has a continuous distribution with p.d.f. f(x)=1/(1+x)² for x>0 and 0 otherwise. Let Y be the amount that the insurance company has to pay on the claim.

(a) Write Y as a function of X, i.e., Y=r(X).

Answer 1

The insurance company's payment, Y, is determined by deducting $100 from the claim amount X, with a maximum cap of $5000. The function Y = r(X) is expressed as Y = min(X - 100, 5000).

Let X be the dollar amount of a claim. The insurance company pays the amount Y, where Y is the amount after deductibles and caps are considered.

Given that the deductible is $100 and the cap is $5000, we can define (Y) as follows:

`\[ Y = \begin{cases} X - 100, & \text{if } X > 100 \text{ and } X \leq 5100 \\ 5000, & \text{if } X > 5100 \\ 0, & \text{if } X \leq 100 \end{cases} \]`

This is because the insurance company pays the amount X - 100 if X is greater than $100 and less than or equal to $5100. If X exceeds $5100, the company pays the maximum of $5000. If X is $100 or less, the company pays nothing.

So, the function Y = r(X) is defined as above.