Independent solution

How to solve this Poisson Distribution question

Setup

Setup

Assign rates to the two Poisson variables and use the fact that each rate is both its mean and its variance.

X∼Poisson⁡(λX),Y∼Poisson⁡(λY)X\sim\operatorname{Poisson}(\lambda_X),\quad Y\sim\operatorname{Poisson}(\lambda_Y)
E[X]=Var⁡(X)=λX,E[Y]=Var⁡(Y)=λY\mathbb{E}[X]=\operatorname{Var}(X)=\lambda_X,\quad \mathbb{E}[Y]=\operatorname{Var}(Y)=\lambda_Y

Model

Model

Translate the mean-difference and variance-ratio information into two equations for the rates.

λX−λY=3\lambda_X-\lambda_Y=3
λX=2λY\lambda_X=2\lambda_Y

Compute

Compute

Solve for the target rate, then apply the Poisson second-moment identity.

2λY−λY=3⟹λY=32\lambda_Y-\lambda_Y=3\quad\Longrightarrow\quad\lambda_Y=3
E[Y2]=Var⁡(Y)+E[Y]2\mathbb{E}[Y^2]=\operatorname{Var}(Y)+\mathbb{E}[Y]^2
E[Y2]=3+32=12\mathbb{E}[Y^2]=3+3^2=12

Answer

Answer

The second raw moment is 12, the value listed under choice D.

E[Y2]=12(D)\boxed{\mathbb{E}[Y^2]=12\quad\text{(D)}}