Editing Weibull distribution (section)

==Properties==

===Density function===

The form of the density function of the Weibull distribution changes drastically with the value of ''k''. For 0 < ''k'' < 1, the density function tends to ∞ as ''x'' approaches zero from above and is strictly decreasing. For ''k'' = 1, the density function tends to 1/''λ'' as ''x'' approaches zero from above and is strictly decreasing. For ''k'' > 1, the density function tends to zero as ''x'' approaches zero from above, increases until its mode and decreases after it. The density function has infinite negative slope at ''x'' = 0 if 0 < ''k'' < 1, infinite positive slope at ''x'' = 0 if 1 < ''k'' < 2 and null slope at ''x'' = 0 if ''k'' > 2. For ''k'' = 1 the density has a finite negative slope at ''x'' = 0. For ''k'' = 2 the density has a finite positive slope at ''x'' = 0. As ''k'' goes to infinity, the Weibull distribution converges to a [[Dirac delta distribution]] centered at ''x'' = λ. Moreover, the skewness and coefficient of variation depend only on the shape parameter. A generalization of the Weibull distribution is the [[hyperbolastic functions|hyperbolastic distribution of type III]].

===Cumulative distribution function===

The [[cumulative distribution function]] for the Weibull distribution is

:<math>F(x;k,\lambda) = 1 - e^{-(x/\lambda)^k}\,</math>

for ''x'' ≥ 0, and ''F''(''x''; ''k''; λ) = 0 for ''x'' < 0.

If ''x'' = λ then ''F''(''x''; ''k''; λ) =&nbsp;1&nbsp;−&nbsp;''e''<sup>−1</sup> ≈&nbsp;0.632 for all values of&nbsp;''k''. Vice versa: at ''F''(''x''; ''k''; ''λ'') = 0.632 the value of&nbsp;''x''&nbsp;≈&nbsp;''λ''.

The quantile (inverse cumulative distribution) function for the Weibull distribution is

:<math>Q(p;k,\lambda) = \lambda(-\ln(1-p))^{1/k}</math>

for 0 ≤ ''p'' < 1.

The [[failure rate]] ''h'' (or hazard function) is given by

:<math> h(x;k,\lambda) = {k \over \lambda} \left({x \over \lambda}\right)^{k-1}.</math>

The [[Mean time between failures]] ''MTBF'' is

:<math> \text{MTBF}(k,\lambda) = \lambda\Gamma(1+1/k).</math>

===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>

===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.

===Minima===
Let <math>X_1, X_2, \ldots, X_n</math> be independent and identically distributed Weibull random variables with scale parameter <math>\lambda</math> and shape parameter <math>k</math>. If the minimum of these <math>n</math> random variables is <math>Z = \min(X_1, X_2, \ldots, X_n)</math>, then the cumulative probability distribution of <math>Z</math> is given by 

:<math>F(z) = 1 - e^{-n(z/\lambda)^k}.</math>

That is, <math>Z</math> will also be Weibull distributed with scale parameter <math>n^{-1/k} \lambda</math> and with shape parameter <math>k</math>.

=== Reparametrization tricks ===
Fix some <math>\alpha > 0</math>. Let <math>(\pi_1, ..., \pi_n)</math> be nonnegative, and not all zero, and let <math>g_1,... , g_n</math> be independent samples of <math>\text{Weibull}(1, \alpha^{-1})</math>, then<ref>{{Cite journal |last1=Balog |first1=Matej |last2=Tripuraneni |first2=Nilesh |last3=Ghahramani |first3=Zoubin |last4=Weller |first4=Adrian |date=2017-07-17 |title=Lost Relatives of the Gumbel Trick |url=https://proceedings.mlr.press/v70/balog17a.html |journal=International Conference on Machine Learning |language=en |publisher=PMLR |pages=371–379|arxiv=1706.04161 }}</ref>

* <math>\arg\min_i (g_i \pi_i^{-\alpha}) \sim \text{Categorical}\left(\frac{\pi_j}{\sum_i \pi_i}\right)_j</math>
* <math>\min_i (g_i \pi_i^{-\alpha}) \sim\text{Weibull}\left( \left(\sum_i \pi_i \right)^{-\alpha}, \alpha^{-1}\right)</math>.

===Shannon entropy===
The [[entropy (information theory)|information entropy]] is given by<ref>{{Cite journal |last1=Cho |first1=Youngseuk |last2=Sun |first2=Hokeun |last3=Lee |first3=Kyeongjun |date=5 January 2015 |title=Estimating the Entropy of a Weibull Distribution under Generalized Progressive Hybrid Censoring |journal=Entropy |language=en |volume=17 |issue=1 |pages=102–122 |doi=10.3390/e17010102 |doi-access=free |bibcode=2015Entrp..17..102C |issn=1099-4300}}</ref>

:<math>
H(\lambda,k) = \gamma\left(1 - \frac{1}{k}\right) + \ln\left(\frac{\lambda}{k}\right) + 1
</math>

where <math>\gamma</math> is the [[Euler–Mascheroni constant]]. The Weibull distribution is the [[maximum entropy distribution]] for a non-negative real random variate with a fixed [[expected value]] of ''x''<sup>''k''</sup> equal to ''&lambda;''<sup>''k''</sup> and a fixed expected value of ln(''x''<sup>''k''</sup>) equal to ln(''&lambda;''<sup>''k''</sup>)&nbsp;−&nbsp;<math>\gamma</math>.

=== Kullback–Leibler divergence ===
The [[Kullback–Leibler divergence]] between two Weibull distributions is given by<ref>{{Cite arXiv |eprint=1310.3713 |class=cs.IT |first1=Christian |last1=Bauckhage |title=Computing the Kullback-Leibler Divergence between two Weibull Distributions |year=2013}}</ref>
: <math>D_\text{KL}( \mathrm{Weib}_1 \parallel \mathrm{Weib}_2) = \log \frac{k_1}{\lambda_1^{k_1}} - \log \frac{k_2}{\lambda_2^{k_2}} + (k_1 - k_2) \left[ \log \lambda_1 - \frac{\gamma}{k_1} \right] + \left(\frac{\lambda_1}{\lambda_2}\right)^{k_2}  \Gamma \left(\frac{k_2}{k_1} + 1 \right) - 1</math>