# Talk:Moment Generating Function of Geometric Distribution

Per the Definition of Geometric Distribution:

$X$ has the geometric distribution with parameter $p$ if and only if:

$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
$\map \Pr {X = k} = \paren {1 - p} p^k$

where 0 < p < 1.

I think we need to edit Moment Generating Function of Geometric Distribution as follows:

From: Let $X$ be a discrete random variable with a geometric distribution with parameter $p$ for some $0 < p \le 1$.

To: Let $X$ be a discrete random variable with a geometric distribution with parameter $p$ for some $0 < p < 1$.

Am I missing something? --Robkahn131 (talk) 13:38, 16 April 2021 (UTC)

Quite right. Good shout. --prime mover (talk) 16:45, 16 April 2021 (UTC)

The definition used on this page $\map \Pr {X = k} = \paren {1 - p}^k p$ is inconsistent with the definitions used on virtually all other pages
Official $\mathsf{Pr} \infty \mathsf{fWiki}$ Definition. Definition of Geometric Distribution
$\map \Pr {X = k} = \paren {1 - p} p^k$
Expectation of Geometric Distribution
Variance of Geometric Distribution
--Robkahn131 (talk) 18:41, 16 April 2021 (UTC)
Take a look at Definition:Geometric Distribution#Note which discusses this.
This page was written by someone obviously using a different source work from the one used for the definition as used on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Please do not automatically assume that if a page says something different from what you find somewhere else that it is automatically wrong. There may be a context in which it is just different. I think this is one of those cases, where both definitions may need to be reconciled. Otherwise someone's going to read this page and say: idiot, everybody knows $\map \Pr {X = k} = \paren {1 - p}^k p$, and change it back to what it was.
The best approach is to document the result for both definitions. --prime mover (talk) 20:18, 16 April 2021 (UTC)
The two formulations have now been presented in parallel. It is trivial to note that if you set $q = 1 - p$, and denote "chance of success" as $p$, then they are isomorphic -- just swap $p$ and $q$. The first one models the number of successes before the first failure, while the second models the number of failures before the first success.
Compare with the shifted geometric distribution which models the number of trials to achieve a success (that is, the first $k - 1$ trials are failures, the $k$th trial is a success).
There is considerable work available to be done to draw all these threads together and present a unified approach that not only removes all confusion, but also documents all different approaches into one at-a-glance thesis. As far as I know this has not actually been done anywhere. All approaches I have seen take one or the other formulation for the geometric distribution and ignores the other. --prime mover (talk) 11:11, 17 April 2021 (UTC)

To calculate the nth moment generating function of $X$ , $\mathsf{Pr} \infty \mathsf{fWiki}$ needs a symbol to represent the Eulerian number <n,k> or A(n,k)
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
$\map \Pr {X = k} = p \paren {1 - p}^k$
The nth moment generating function of $X$ is given by:
$\ds \map { {M_X}^{\paren n} } t = \dfrac {p \paren {1 - p } e^t} {\paren {1 - \paren {1 - p} e^t}^{n + 1} } \sum_{k \mathop = 0}^{n-1} A\paren {n, k} \paren {1 - p}^k e^k$
https://mathworld.wolfram.com/EulerianNumber.html
https://en.wikipedia.org/wiki/Eulerian_number
Is the Eulerian number on this site and I'm just not seeing it? Thoughts on the official notation? --Robkahn131 (talk) 16:16, 22 April 2021 (UTC)
Nope, I believe they're not, although they might be touched on in Knuth TAOCP. Nice research, I had a feeling they were in the frame somewhere here. Be good to get this documented as I'm not sure this use of them is readily available on the web. --prime mover (talk) 22:28, 22 April 2021 (UTC)