Independent solution

How to solve this Independence question

Setup

Setup

Write the payment as the product of the independent exponential amount and the zero-one payment indicator.

Y=ZX,Pr⁡(Z=1)=0.45Y=ZX,\qquad \Pr(Z=1)=0.45

Model

Model

Recover the first two raw moments of the exponential amount, then use Z squared equals Z.

E⁡[X]=8,Var⁡(X)=82=64\operatorname{E}[X]=8,\qquad \operatorname{Var}(X)=8^2=64
E⁡[X2]=64+82=128\operatorname{E}[X^2]=64+8^2=128
Z2=ZZ^2=Z

Compute

Compute

Independence factors the product moments.

E⁡[Y]=E⁡[Z]E⁡[X]=(0.45)(8)=3.6\operatorname{E}[Y]=\operatorname{E}[Z]\operatorname{E}[X]=(0.45)(8)=3.6
E⁡[Y2]=E⁡[Z2]E⁡[X2]=(0.45)(128)=57.6\operatorname{E}[Y^2]=\operatorname{E}[Z^2]\operatorname{E}[X^2]=(0.45)(128)=57.6
Var⁡(Y)=57.6−3.62=44.64\operatorname{Var}(Y)=57.6-3.6^2=44.64

Answer

Answer

The payment variance is 44.64.

44.64(E)\boxed{44.64\quad\text{(E)}}