Independent solution

How to solve this Poisson Distribution question

Setup

Setup

Use the ratio of consecutive Poisson probabilities to characterize when neighboring counts have equal maximum probability.

pk=e−λλkk!,pkpk−1=λkp_k=e^{-\lambda}\frac{\lambda^k}{k!},\qquad \frac{p_k}{p_{k-1}}=\frac{\lambda}{k}

Model

Model

For the distribution whose two modes are 2 and 3, those two masses must be equal.

p3p2=λlow3=1\frac{p_3}{p_2}=\frac{\lambda_{\mathrm{low}}}{3}=1
λlow=3\lambda_{\mathrm{low}}=3

Compute

Compute

Apply the four-to-one mean relationship, recalling that a Poisson mean equals its rate. For an integer rate n, the equal largest masses occur at n-1 and n.

λhigh=4λlow=4(3)=12\lambda_{\mathrm{high}}=4\lambda_{\mathrm{low}}=4(3)=12
Modes⁡(Poisson⁡(12))={11,12}\operatorname{Modes}(\operatorname{Poisson}(12))=\{11,12\}

Answer

Answer

The two modal counts are 11 and 12, which are listed together under choice D.

11 and 12(D)\boxed{11\text{ and }12\quad\text{(D)}}