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.. _continuous-invgamma:
Inverted Gamma Distribution
===========================
Special case of the generalized Gamma distribution with :math:`c=-1` and :math:`a>0` and support :math:`x\geq0`.
.. math::
:nowrap:
\begin{eqnarray*} f\left(x;a\right) & = & \frac{x^{-a-1}}{\Gamma\left(a\right)}\exp\left(-\frac{1}{x}\right)\\
F\left(x;a\right) & = & \frac{\Gamma\left(a,\frac{1}{x}\right)}{\Gamma\left(a\right)}\\
G\left(q;a\right) & = & \left\{ \Gamma^{-1}\left(a,\Gamma\left(a\right)q\right)\right\} ^{-1}\end{eqnarray*}
.. math::
\mu_{n}^{\prime}=\frac{\Gamma\left(a-n\right)}{\Gamma\left(a\right)}\quad a>n
.. math::
:nowrap:
\begin{eqnarray*} \mu & = & \frac{1}{a-1}\quad a>1\\
\mu_{2} & = & \frac{1}{\left(a-2\right)\left(a-1\right)}-\mu^{2}\quad a>2\\
\gamma_{1} & = & \frac{\frac{1}{\left(a-3\right)\left(a-2\right)\left(a-1\right)}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\gamma_{2} & = & \frac{\frac{1}{\left(a-4\right)\left(a-3\right)\left(a-2\right)\left(a-1\right)}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
.. math::
m_{d}=\frac{1}{a+1}
.. math::
h\left[X\right]=a-\left(a+1\right)\psi\left(a\right)+\log\Gamma\left(a\right).
where :math:`\Psi` is the digamma function :math:`\psi(z) = \frac{d}{dz} \log(\Gamma(z))`.
Implementation: `scipy.stats.invgamma`
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