Independent solution

How to solve this Mixture Distributions question

Setup

Setup

Let I indicate that a positive payment occurs, and let Z be the positive exponential amount.

Pr⁡(I=1)=0.25,E[Z]=8,E[Z2]=2(82)=128\Pr(I=1)=0.25,\qquad E[Z]=8,\qquad E[Z^2]=2(8^2)=128
Y=IZY=IZ

Model

Model

Average the first two conditional moments across the zero and positive components.

E[Y]=0.25(8)=2E[Y]=0.25(8)=2
E[Y2]=0.25(128)=32E[Y^2]=0.25(128)=32

Compute

Compute

Convert the raw second moment to variance.

Var⁡(Y)=E[Y2]−E[Y]2\operatorname{Var}(Y)=E[Y^2]-E[Y]^2
Var⁡(Y)=32−22=28\operatorname{Var}(Y)=32-2^2=28

Answer

Answer

The payment variance is 28.

28(D)\boxed{28\quad\text{(D)}}