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 \(k = 1,2,...\).
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.