Editing Weibull distribution (section)

===Moments===
The [[moment generating function]] of the [[logarithm]] of a Weibull distributed [[random variable]] is given by<ref name=JKB>{{harvnb|Johnson|Kotz|Balakrishnan|1994}}</ref>

:<math>\operatorname E\left[e^{t\log X}\right] = \lambda^t\Gamma\left(\frac{t}{k}+1\right)</math>

where {{math|Γ}} is the [[gamma function]]. Similarly, the [[characteristic function (probability theory)|characteristic function]] of log ''X'' is given by

:<math>\operatorname E\left[e^{it\log X}\right] = \lambda^{it}\Gamma\left(\frac{it}{k}+1\right).</math>

In particular, the ''n''th [[raw moment]] of ''X'' is given by

:<math>m_n = \lambda^n \Gamma\left(1+\frac{n}{k}\right).</math>

The [[mean]] and [[variance]] of a Weibull [[random variable]] can be expressed as

:<math>\operatorname{E}(X) = \lambda \Gamma\left(1+\frac{1}{k}\right)\,</math>

and

:<math>\operatorname{var}(X) = \lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right]\,.</math>

The skewness is given by

:<math>\gamma_1=\frac{2\Gamma_1^3-3\Gamma_1\Gamma_2+ \Gamma_3 }{[\Gamma_2-\Gamma_1^2]^{3/2}}</math>

where <math>\Gamma_i=\Gamma(1+i/k)</math>, which may also be written as  

:<math>\gamma_1=\frac{\Gamma\left(1+\frac{3}{k}\right)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3}</math>

where the mean is denoted by {{math|μ}} and the standard deviation is denoted by {{math|σ}}.

The excess [[kurtosis]] is given by

:<math>\gamma_2=\frac{-6\Gamma_1^4+12\Gamma_1^2\Gamma_2-3\Gamma_2^2-4\Gamma_1 \Gamma_3 +\Gamma_4}{[\Gamma_2-\Gamma_1^2]^2}</math>

where <math>\Gamma_i=\Gamma(1+i/k)</math>. The kurtosis excess may also be written as:

:<math>\gamma_2=\frac{\lambda^4\Gamma(1+\frac{4}{k})-4\gamma_1\sigma^3\mu-6\mu^2\sigma^2-\mu^4}{\sigma^4}-3.</math>