Independent solution

How to solve this Exponential Distribution question

Setup

Setup

Write the payment as the positive part above the deductible and denote the exceedance probability by p.

Y=(X−d)+Y=(X-d)_+
p=Pr⁡(X>d)=e−d/10p=\Pr(X>d)=e^{-d/10}

Model

Model

Given a positive payment, exponential memorylessness makes the excess exponential with the original mean 10.

E⁡[Y∣Y>0]=10\operatorname{E}[Y\mid Y>0]=10
E⁡[Y2∣Y>0]=2(102)=200\operatorname{E}[Y^2\mid Y>0]=2(10^2)=200

Compute

Compute

Multiply the conditional moments by the positive-payment probability and subtract the squared unconditional mean.

E⁡[Y]=10e−d/10,E⁡[Y2]=200e−d/10\operatorname{E}[Y]=10e^{-d/10},\qquad \operatorname{E}[Y^2]=200e^{-d/10}
Var⁡(Y)=200e−d/10−100e−d/5\operatorname{Var}(Y)=200e^{-d/10}-100e^{-d/5}
Var⁡(Y)=100(2e−d/10−e−d/5)\operatorname{Var}(Y)=100\left(2e^{-d/10}-e^{-d/5}\right)

Answer

Answer

The variance matches the expression in choice D.

100(2e−d/10−e−d/5)(D)\boxed{100\left(2e^{-d/10}-e^{-d/5}\right)\quad\text{(D)}}