Independent solution

How to solve this Sums of Independent Random Variables question

Answer in brief

Independence lets the second component inherit mean 0.8 and variance 1.35 from the stated totals. Its coefficient of variation is sqrt(1.35)/0.8 = 1.45237, so the answer is E.

Setup

Setup

Recover the mean of the unobserved component by subtracting component means.

E[Z]=E[X]−E[Y]=2.00−1.20=0.80E[Z]=E[X]-E[Y]=2.00-1.20=0.80

Model

Model

For independent components, the variance of their sum is the sum of their variances.

Var⁡(Z)=Var⁡(X)−Var⁡(Y)\operatorname{Var}(Z)=\operatorname{Var}(X)-\operatorname{Var}(Y)
Var⁡(Z)=2.25−0.90=1.35\operatorname{Var}(Z)=2.25-0.90=1.35

Compute

Compute

Divide the component standard deviation by its mean.

CV⁡(Z)=1.350.80=1.452368754…\operatorname{CV}(Z)=\frac{\sqrt{1.35}}{0.80}=1.452368754\ldots

Answer

Answer

The coefficient of variation rounds to 1.45.

1.45(E)\boxed{1.45\quad\text{(E)}}