How can I differentiate a Poisson distribution to find the maximum that way?

Question

I want to find the maximum to a Poisson distribution using calculus. How can I do this? I want to do this algebraically.

Answer 1

The Poisson distribution is only defined for $k\in\mathbb{N}$ and therefore can't be differentiated.

Answer 2

One way to show the maximum is using the following method: $\sum_{x=0}^{\infty}e^{−µ}\frac{µ^x}{x!}$ = $e^{−µ}\sum_{x=0}^{\infty}\frac{µ^x}{x!}$ = $e^µe^{-µ}$ = 1

Answer 3

Hint: The ratio of two consecutive Poisson probabilities is $\frac{p_{n+1}}{p_n}=\frac{e^{-\mu}\mu^{n+1}/(n+1)!}{e^{-\mu}\mu^{n}/n!}=\frac{\mu}{n+1}$. For what $n$ is this ratio greater/less than one?