doc/source/tutorial/stats/discrete_yulesimon.rst
.. _discrete-yulesimon:
A Yule-Simon random variable with parameter :math:\alpha>0
can be represented as a mixture of
exponential random variates. To see this write :math:W as an exponential
random variate with rate :math:\rho and a Geometric random variate :math:K
with probability :math:1-exp(-W) then :math:K marginally has a Yule-Simon
distribution. The latent variable representation described above is used for
random variate generation.
.. math:: :nowrap:
\begin{eqnarray*}
p \left( k; \alpha \right) & = & \alpha \frac{\Gamma\left(k\right)\Gamma\left(\alpha + 1\right)}{\Gamma\left(k+\alpha+1\right)} \\
F \left( k; \alpha \right) & = & 1 - \frac{ k \Gamma\left(k\right)\Gamma\left(\alpha + 1\right)}{\Gamma\left(k+\alpha+1\right)}
\end{eqnarray*}
for :math:k = 1,2,....
Now
.. math:: :nowrap:
\begin{eqnarray*} \mu & = & \frac{\alpha}{\alpha-1}\\
\mu_{2} & = & \frac{\alpha^2}{\left(\alpha-1\right)^2\left( \alpha - 2 \right)}\\
\gamma_{1} & = & \frac{ \sqrt{\left( \alpha - 2 \right)} \left( \alpha + 1 \right)^2}{ \alpha \left( \alpha - 3 \right)}\\
\gamma_{2} & = & \frac{ \left(\alpha + 3\right) + \left(\alpha^3 - 49\alpha - 22\right)}{\alpha \left(\alpha - 4\right)\left(\alpha - 3 \right) }
\end{eqnarray*}
for :math:\alpha>1 otherwise the mean is infinite and the variance does not exist.
For the variance, :math:\alpha>2 otherwise the variance does not exist.
Similarly, for the skewness and
kurtosis to be finite, :math:\alpha>3 and :math:\alpha>4 respectively.
Implementation: scipy.stats.yulesimon