Maximum Value of a Poisson Distribution

Question

Let $X$ be a Poisson random variable with parameter $\lambda$. What is the most likely outcome of $X$?

Please help me solve this exercise. Thanks in advance! :)

Answer 1

For $n\ge 0$, $$P(X=n)=e^{-\lambda}\frac{\lambda^n}{n!}$$ In particular $P(X=n)=\frac{\lambda}{n}P(X=n-1)$. So the sequence $P(X=n)$ grows for a while, then it's eventualy decreasing. And the reason/measure of this is...