Talk:Poisson distribution

Ceiling function
The same way the symbols used for the floor function are explained (in the Infobox-cdf), shouldn't the symbols for the ceiling function be also defined (appear in the "Infobox-mode" of the function)? 193.136.147.158 (talk) 13:48, 23 October 2023 (UTC)

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:
 * $$F(k;\lambda) = \Pr(X \le k) = e^{- \lambda} \sum_{i=0}^{\lfloor k \rfloor} \frac{\lambda^i}{i!},$$
 * 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 $$\Gamma$$ or the regularized gamma function $$Q$$, as follows:
 * $$F(k;\lambda) = \frac{\Gamma(\lfloor k + 1 \rfloor, \lambda}{\lfloor k \rfloor!} = Q(\lfloor k + 1 \rfloor, \lambda).$$
 * FynnFreyer (talk) 19:24, 21 November 2023 (UTC)
 * FynnFreyer (talk) 19:24, 21 November 2023 (UTC)

Other Properties - Mitzenmacher
Source does not support this statement in the article: and $$\Pr(Y \leq E[Y]) \geq \frac{1}{2}.$$

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:
 * {| class="wikitable"

! Value of $$\mu$$ !! $$\mathbb{P}(Y \leq \mu) = \sum_{k = 0}^\mu \frac{e^{-\mu} \mu^k}{k!} $$ !! 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).
 * 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 $$(Y_\mu - \mu)_{\mu \in \mathbb{N}}$$ 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)