Independent solution

How to solve this Poisson Distribution question

Setup

Setup

Write the Poisson mean and variance in terms of its rate parameter.

E[X]=λ,Var⁡(X)=λ\mathbb{E}[X]=\lambda,\qquad \operatorname{Var}(X)=\lambda

Model

Model

Connect the given second moment to the rate through the variance identity.

E[X2]=Var⁡(X)+E[X]2=λ+λ2\mathbb{E}[X^2]=\operatorname{Var}(X)+\mathbb{E}[X]^2=\lambda+\lambda^2
λ2+λ=6\lambda^2+\lambda=6

Compute

Compute

Only the nonnegative root is admissible for a Poisson rate. Compute the requested tail by subtracting the first three masses.

(λ−2)(λ+3)=0⟹λ=2(\lambda-2)(\lambda+3)=0\quad\Longrightarrow\quad\lambda=2
Pr⁡(X≥3)=1−∑k=02e−22kk!\Pr(X\ge 3)=1-\sum_{k=0}^{2}e^{-2}\frac{2^k}{k!}
Pr⁡(X≥3)=1−5e−2=0.3233235838…\Pr(X\ge 3)=1-5e^{-2}=0.3233235838\ldots

Answer

Answer

The computed value rounds to the value listed under choice C.

Pr⁡(X≥3)≈0.323324(C)\boxed{\Pr(X\ge 3)\approx 0.323324\quad\text{(C)}}