How can I differentiate a Poisson distribution to find the maximum that way? I want to find the maximum to a Poisson distribution using calculus. How can I do this? I want to do this algebraically.
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?
The Poisson distribution is only defined for $k\in\mathbb{N}$ and therefore can't be differentiated.