Independent solution

How to solve this Limited Expected Value question

Setup

Setup

Let the payment be the loss above the deductible, limited to the stated maximum. The payment exceeds a level only when the original loss exceeds that level plus the deductible.

Y=min⁡((X−1)+,5),Pr⁡(X>x)=e−x/2Y=\min((X-1)_+,5),\quad \Pr(X>x)=e^{-x/2}

Model

Model

Use the survival-integral identity for a nonnegative limited payment. Shifting by the deductible changes the integration interval to losses from one through six.

E[Y]=∫05Pr⁡(Y>y) dy=∫16e−x/2 dxE[Y]=\int_0^5\Pr(Y>y)\,dy=\int_1^6e^{-x/2}\,dx

Compute

Compute

Integrating the exponential survival function over that interval gives approximately 0.922701.

E[Y]=2e−1/2−2e−3=0.922701E[Y]=2e^{-1/2}-2e^{-3}=0.922701

Answer

Answer

The expected claim payment is approximately 0.922701, so choice C is correct.

2e−1/2−2e−3(C)\boxed{2e^{-1/2}-2e^{-3}\quad\text{(C)}}