Talk:Poisson distribution

Adding a section on the CDF

Would it make sense to add a subsection for the CDF under the definitions, like there is for the Binomial distribution? FynnFreyer (talk) 19:07, 21 November 2023 (UTC)

Modelled after the linked section it could look like this:

The cumulative distribution function can be expressed as:

$F(k;\lambda )=\mathrm{Pr}(X\le k)=e^{-\lambda }\sum _{i=0}^{\lfloor k\rfloor }\frac{{\lambda }^i}{i!},$

where $\lfloor k\rfloor$ is the "floor" under k, i.e. the greatest integer less than or equal to k, and $!$ is the factorial function.

It can also be represented in terms of the upper incomplete gamma function $\mathrm{\Gamma }$ or the regularized gamma function $Q$, as follows:

$F(k;\lambda )=\frac{\mathrm{\Gamma }(\lfloor k+1\rfloor ,\lambda }{\lfloor k\rfloor !}=Q(\lfloor k+1\rfloor ,\lambda ).$

FynnFreyer (talk) 19:24, 21 November 2023 (UTC)

Other Properties - Mitzenmacher

Source does not support this statement in the article:

and $\mathrm{Pr}(Y\le E[Y])\ge \frac{1}{2}.$

See https://imgur.com/a/3lE0VDa

2A02:1811:351E:AF00:2966:4372:24C8:154B (talk) — Preceding undated comment added 23:58, 12 January 2024 (UTC)

You are right: there is a typo in the source. However the statement on Wikipedia is correct:

Value of $\mu$	$\mathbb{\mathbb{P}}(Y\le \mu )=\sum _{k=0}^{\mu }\frac{e^{-\mu }{\mu }^k}{k!}$	Numerical approx.
1	2/e	0.73575...
2	5/e2	0.67667...
3	13/e3	0.64723...
4	—	0.62883...
5	—	0.61596...

More values can be obtained, e.g, with the following Python function

f = lambda n : sum(n**k * math.exp(-n) / math.factorial(k) for k in range(n + 1))

(note that this is poorly implemented, and that it overflows for μ ≥ 144).

In view of this, it's pretty clear that the mistake in the source is a typo rather than an actual mathematical error. Still, it's a problem... Especially since I wouldn't know where to find a source for this kind of statement. It's not too hard to see that the statement should be true for large $\mu$ (e.g, because the variables $Y_{\mu }$ can be coupled in such a way that $(Y_{\mu }-\mu {)}_{\mu \in \mathbb{\mathbb{N}}}$ is a random walk whose increments are centered and have variance 1), but even if someone provides a proof here, it might be considered original research.

As far as I'm concerned:

• the fact that there is a mistake is not a huge problem, since it's clearly a typo; but I understand that some people might disagree;
• the fact that there is no proof in the source is a bigger problem;
• I think the statement is cool, but it's relevance is actually not so clear.

Malparti (talk) 18:57, 13 January 2024 (UTC)

Add a simple introduction

This article is hard to read unless you already know what a Poisson distribution is, and that is unnecessary.

It would help to start out with a simple introduction of the term Poisson_process. Perhaps add an illustration to help the reader. Subsequently use this section to define the Poisson distribution. — Preceding unsigned comment added by 89.23.239.207 (talk) 13:28, 23 September 2024 (UTC)