Independent solution

How to solve this Poisson Distribution question

Setup

Setup

Let N be the annual count and express the company's cost after the insurance layer.

N∼Poisson⁡(2)N\sim\operatorname{Poisson}(2)
C=100min⁡(N,4)C=100\min(N,4)

Model

Model

For a nonnegative integer count, the expected capped count is the sum of its first four survival probabilities.

E⁡[min⁡(N,4)]=∑j=14Pr⁡(N≥j)\operatorname{E}[\min(N,4)]=\sum_{j=1}^{4}\Pr(N\ge j)

Compute

Compute

Evaluate the Poisson tails through the attachment count.

E⁡[C]=100[Pr⁡(N≥1)+Pr⁡(N≥2)+Pr⁡(N≥3)+Pr⁡(N≥4)]\operatorname{E}[C]=100\left[\Pr(N\ge1)+\Pr(N\ge2)+\Pr(N\ge3)+\Pr(N\ge4)\right]
E⁡[C]=100[4−463e−2]\operatorname{E}[C]=100\left[4-\frac{46}{3}e^{-2}\right]
E⁡[C]=400−46003e−2=192.4858990\operatorname{E}[C]=400-\frac{4600}{3}e^{-2}=192.4858990

Answer

Answer

The expected retained annual cost rounds to 192.

192(C)\boxed{192\quad\text{(C)}}