Graph the describes the peaks of a Poisson Distribution

Question

Probably not a very smart question - barely know any of the more interesting parts of probability theory - but I noticed that the peaks of Poisson curves form what looks like a kind of logarithmic curve. Can't find anything which talks about what curve models the points, but it feels like it should be pretty easy to answer. Someone help me out? Pic of the curve plotted vs 1/(e^2log(x)) Seems close?

Answer 1

The mode of a distribution is the place where the PMF has the highest value (most probable).

The mode of a Poisson distribution with parameter $\lambda$ ( i.e. PMF $\lambda^k e^{- \lambda}/k!$ for $k \in \mathbb{Z}^+$ is $\lfloor \lambda \rfloor$ if $\lambda$ is not an integer, and is $\lambda$ or $\lambda-1$ if it is an integer (these two have equal PMF).

Lets stick with $\lambda$ being an integer (and the results are similar for non-integer $\lambda$). In this case, the probability of the mode is $\frac{\lambda^\lambda e^{-\lambda}}{\lambda!}$, and you want to look at the plot of the probability of the mode versus $\lambda$. Stirling's approximation says that $n!$ is approximately $\sqrt{2 \pi n} \left( \frac{n}{e} \right)^n$ for large $n$, so we can approximate $\frac{\lambda^\lambda}{\lambda!}$ by something like $e^\lambda/\sqrt{2 \pi \lambda}$.

So, the probability of the mode is approximately $1/ \sqrt{ 2 \pi \lambda}$. You should try plotting $y = \frac{1}{\sqrt{2 \pi x}}$ instead.