Editing Incomplete gamma function (section)

== Regularized gamma functions and Poisson random variables ==

Two related functions are the regularized gamma functions:
<math display="block">\begin{align}
P(s,x) &= \frac{\gamma(s,x)}{\Gamma(s)}, \\[1ex]
Q(s,x) &= \frac{\Gamma(s,x)}{\Gamma(s)} = 1 - P(s,x).
\end{align}</math>
<math>P(s,x)</math> is the [[cumulative distribution function]] for [[Gamma distribution|gamma random variables]] with [[shape parameter]] <math>s</math> and [[scale parameter]] 1.

When <math>s</math> is an integer, <math>Q(s+1, \lambda)</math> is the cumulative distribution function for [[Poisson random variable]]s: If <math>X</math> is a <math>\mathrm{Poi}(\lambda)</math> random variable then
<math display="block"> \Pr(X \leq s) = \sum_{i \leq s} e^{-\lambda} \frac{\lambda^i}{i!} = \frac{\Gamma(s+1,\lambda)}{\Gamma(s+1)} = Q(s+1,\lambda).</math>

This formula can be derived by repeated integration by parts.

In the context of the [[stable count distribution]], the <math> s </math> parameter can be regarded as inverse of Lévy's stability parameter <math> \alpha</math>: 
<math display="block"> Q(s,x) = \int_0^\infty e^{\left( -{x^s}/{\nu} \right)} \, \mathfrak{N}_{{1}/{s}}\left(\nu\right)  \, d\nu , \quad (s > 1)</math>
where <math>\mathfrak{N}_{\alpha}(\nu)</math> is a standard stable count distribution of shape <math> \alpha = 1/s < 1</math>.

<math>P(s,x)</math> and <math>Q(s, x)</math> are implemented as <code>gammainc</code><ref>{{Cite web|url=https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammainc.html#scipy.special.gammainc|title=scipy.special.gammainc — SciPy v1.11.4 Manual|website=docs.scipy.org}}</ref> and <code>gammaincc</code><ref>{{Cite web|url=https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammaincc.html|title=scipy.special.gammaincc — SciPy v1.11.4 Manual|website=docs.scipy.org}}</ref> in [[scipy]].