Editing Weibull distribution (section)

===Moment generating function===
A variety of expressions are available for the moment generating function of ''X'' itself. As a [[power series]], since the raw moments are already known, one has

:<math>\operatorname E\left[e^{tX}\right] = \sum_{n=0}^\infty \frac{t^n\lambda^n}{n!} \Gamma\left(1+\frac{n}{k}\right).</math>

Alternatively, one can attempt to deal directly with the integral

:<math>\operatorname E\left[e^{tX}\right] = \int_0^\infty e^{tx} \frac k \lambda \left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}\,dx.</math>

If the parameter ''k'' is assumed to be a rational number, expressed as ''k'' = ''p''/''q'' where ''p'' and ''q'' are integers, then this integral can be evaluated analytically.{{efn |See {{harv|Cheng|Tellambura|Beaulieu|2004}} for the case when ''k'' is an integer, and {{harv|Sagias|Karagiannidis|2005}} for the rational case.}} With ''t'' replaced by −''t'', one finds

:<math> \operatorname E\left[e^{-tX}\right] = \frac1{ \lambda^k\, t^k} \, \frac{ p^k \, \sqrt{q/p}} {(\sqrt{2 \pi})^{q+p-2}} \, G_{p,q}^{\,q,p} \!\left( \left. \begin{matrix} \frac{1-k}{p}, \frac{2-k}{p}, \dots, \frac{p-k}{p} \\ \frac{0}{q}, \frac{1}{q}, \dots, \frac{q-1}{q} \end{matrix} \; \right| \, \frac {p^p} {\left( q \, \lambda^k \, t^k \right)^q} \right) </math>
where ''G'' is the [[Meijer G-function]].

The [[characteristic function (probability theory)|characteristic function]] has also been obtained by {{harvtxt|Muraleedharan|Rao|Kurup|Nair|2007}}. The [[characteristic function (probability theory)|characteristic function]] and [[moment generating function]] of 3-parameter Weibull distribution have also been derived by {{harvtxt|Muraleedharan|Soares|2014}} by a direct approach.