Independent solution

How to solve this Poisson Tail Probability question

Setup

Setup

Let M be the Poisson count with unknown rate λ and write its probability mass function.

Pr⁡(M=k)=e−λλkk!\Pr(M=k)=e^{-\lambda}\frac{\lambda^k}{k!}

Model

Model

For adjacent Poisson masses, their ratio removes the common exponential factor and most powers.

Pr⁡(M=4)Pr⁡(M=3)=λ4\frac{\Pr(M=4)}{\Pr(M=3)}=\frac{\lambda}{4}
Pr⁡(M=3)=Pr⁡(M=4)⟹λ4=1\Pr(M=3)=\Pr(M=4)\quad\Longrightarrow\quad\frac{\lambda}{4}=1

Compute

Compute

Substitute the identified rate and calculate the requested upper tail by complement.

λ=4\lambda=4
Pr⁡(M≥3)=1−Pr⁡(M≤2)\Pr(M\ge3)=1-\Pr(M\le2)
Pr⁡(M≥3)=1−e−4(1+4+422)=1−13e−4=0.7618966944…\Pr(M\ge3)=1-e^{-4}\left(1+4+\frac{4^2}{2}\right)=1-13e^{-4}=0.7618966944\ldots

Answer

Answer

The tail probability rounds to the value shown in choice E.

Pr⁡(M≥3)≈0.762(E)\boxed{\Pr(M\ge3)\approx0.762\quad\text{(E)}}