if I have a Poisson random variable $X$, how do I find the constant 'k' that makes $P(X=k)$ be max?

Question

well the book suggests to compare with $\mathbb{P}[X=k-1]$, I notice that, when $k$ is equal to the parameter $\lambda$ we have $\mathbb{P}[X=\lambda]={P}[X=\lambda-1]$, $\mathbb{P}[X= \lambda]$ and it's the greatest probability since the expectation of a Poisson distribution is equal to $\lambda$. If I'm right when I say that, how can I state it more properly?

Answer 1

As the hint suggested, we find $\frac{\Pr(X=k)}{\Pr(X=k-1)}$. After a little simplification this turns out to be $\frac{\lambda}{k}$. Thus roughly speaking $\Pr(X=x)$ is increasing while $x\lt \lambda$, and then decreasing. We next make this more precise.

If $\lambda\lt 1$, then $\Pr(X=k)\gt \Pr(X=k-1)$ for all $k\ge 1$. This means that the maximum probability is reached at $0$.

If $\lambda$ is a positive integer, then $\Pr(X=k)=\Pr(X=k-1)$ if $k=\lambda$. So the maximum is reached at two places, $x=\lambda-1$ and $x=\lambda$.

If $\lambda \gt 1$ is not an integer, then the probabilities are increasing until $x=\lfloor \lambda\rfloor$ and then decreasing, so the maximum is reached at $x=\lfloor x\rfloor$.