Independent solution

How to solve this Poisson Distribution question

Setup

Setup

Represent each location-month count separately. Every component is Poisson, so its variance equals its mean.

Var⁡(Xm)=15,Var⁡(Ym)=30\operatorname{Var}(X_m)=15,\qquad \operatorname{Var}(Y_m)=30
m=1,2,3m=1,2,3

Model

Model

The six counts are mutually independent, so the variance of their sum is the sum of their variances.

T=∑m=13(Xm+Ym)T=\sum_{m=1}^{3}(X_m+Y_m)
Var⁡(T)=∑m=13(Var⁡(Xm)+Var⁡(Ym))\operatorname{Var}(T)=\sum_{m=1}^{3}\left(\operatorname{Var}(X_m)+\operatorname{Var}(Y_m)\right)

Compute

Compute

Each month contributes variance 45, and there are three independent months.

Var⁡(T)=3(15+30)=3(45)=135\operatorname{Var}(T)=3(15+30)=3(45)=135

Answer

Answer

The variance of the full three-month total is 135.

Var⁡(T)=135(B)\boxed{\operatorname{Var}(T)=135\quad\text{(B)}}