# Yule-Simon Distribution¶

A Yule-Simon random variable with parameter $$\alpha>0$$ can be represented as a mixture of exponential random variates. To see this write $$W$$ as an exponential random variate with rate $$\rho$$ and a Geometric random variate $$K$$ with probability $$1-exp(-W)$$ then $$K$$ marginally has a Yule-Simon distribution. The latent variable representation described above is used for random variate generation.

\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 $$k = 1,2,...$$.

Now

\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 $$\alpha>1$$ otherwise the mean is infinite and the variance does not exist. For the variance, $$\alpha>2$$ otherwise the variance does not exist. Similarly, for the skewness and kurtosis to be finite, $$\alpha>3$$ and $$\alpha>4$$ respectively.

Implementation: scipy.stats.yulesimon