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.

for $k=1,2,...$.

Now

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