Independent solution

How to solve this Sums of Independent Random Variables question

Answer in brief

The independent components have combined mean 250 and combined standard deviation sqrt(40^2+30^2)=50. Dividing 50 by 250 gives coefficient of variation 0.20, so choice A is correct.

Setup

Setup

Add the two component means to obtain the expected combined loss.

E[S]=100+150=250E[S]=100+150=250

Model

Model

Independence makes the component variances additive.

Var⁡(S)=402+302=2500\operatorname{Var}(S)=40^2+30^2=2500
SD⁡(S)=50\operatorname{SD}(S)=50

Compute

Compute

Form the dimensionless ratio of standard deviation to mean.

CV⁡(S)=SD⁡(S)E[S]\operatorname{CV}(S)=\frac{\operatorname{SD}(S)}{E[S]}
CV⁡(S)=50250=0.20\operatorname{CV}(S)=\frac{50}{250}=0.20

Answer

Answer

The combined coefficient of variation is 0.20.

0.20(A)\boxed{0.20\quad\text{(A)}}