Talk:Moment Generating Function of Geometric Distribution

Per the :

$X$ has the geometric distribution with parameter $p$ :
 * $\map X \Omega = \set {0, 1, 2, \ldots} = \N$
 * $\map \Pr {X = k} = \paren {1 - p} p^k$

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 Definition.
 * $\map \Pr {X = k} = \paren {1 - p} p^k$


 * Expectation of Geometric Distribution
 * Variance of Geometric Distribution
 * --Robkahn131 (talk) 18:41, 16 April 2021 (UTC)


 * Take a look at Definition:Geometric Distribution 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.


 * 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 $X$, needs a symbol to represent the Eulerian number  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)