.. _continuous-gompertz:

Gompertz (Truncated Gumbel) Distribution

For :math:x\geq0 and :math:c>0 . In JKB the two shape parameters :math:b,a are reduced to the single shape-parameter :math:c=b/a . As :math:a is just a scale parameter when :math:a\neq0 . If :math:a=0, the distribution reduces to the exponential distribution scaled by :math:1/b. Thus, the standard form is given as

.. math:: :nowrap:

\begin{eqnarray*} f\left(x;c\right) & = & ce^{x}\exp\left(-c\left(e^{x}-1\right)\right)\\
F\left(x;c\right) & = & 1-\exp\left(-c\left(e^{x}-1\right)\right)\\
G\left(q;c\right) & = & \log\left(1-\frac{1}{c}\log\left(1-q\right)\right)\end{eqnarray*}

.. math::

 h\left[X\right]=1-\log\left(c\right)-e^{c}\mathrm{Ei}\left(1,c\right),

where

.. math::

 \mathrm{Ei}\left(n,x\right)=\int_{1}^{\infty}t^{-n}\exp\left(-xt\right)dt

Implementation: scipy.stats.gompertz