Independent solution

How to solve this Variance question

Setup

Setup

Aggregate the rainfall intervals that produce each distinct benefit.

Pr⁡(B=0)=0.70\Pr(B=0)=0.70
Pr⁡(B=5000,10000,15000,18000)=(0.11,0.08,0.07,0.04)\Pr(B=5000,10000,15000,18000)=(0.11,0.08,0.07,0.04)

Model

Model

Calculate the first two moments of the benefit distribution.

E⁡[B]=5000(0.11)+10000(0.08)+15000(0.07)+18000(0.04)=3120\operatorname{E}[B]=5000(0.11)+10000(0.08)+15000(0.07)+18000(0.04)=3120
E⁡[B2]=39,460,000\operatorname{E}[B^2]=39{,}460{,}000

Compute

Compute

Subtract the squared mean from the second moment and take the square root.

Var⁡(B)=39,460,000−(3120)2=29,725,600\operatorname{Var}(B)=39{,}460{,}000-(3120)^2=29{,}725{,}600
SD⁡(B)=29,725,600=5452.118854…\operatorname{SD}(B)=\sqrt{29{,}725{,}600}=5452.118854\ldots

Answer

Answer

The benefit standard deviation rounds to 5452.

5452(D)\boxed{5452\quad\text{(D)}}