Maximum value with X poisson($\lambda$) distributed

Question

Let $X$ be poisson$(\lambda)$ distributed. For which $k \in N$ is $P[X = k]$ maximal?

Do you need to solve it with the derivative? Because then you need the derivative of $k!$. Or could someone tell me how to do this?

Answer 1

Hint:

Have a look at the function $f$ on positive integers prescribed by:$$k\mapsto \frac{P(X=k)}{P(X=k-1)}=\frac{\lambda}k$$

Be aware of the fact that $$P(X=k)>P(X=k-1)\iff f(k)>1$$ Further the function is monotically decreasing.