Probability mass function
Yule–Simon PMF on a log-log scale. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Yule–Simon CMF. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
The probability mass function (pmf) of the Yule–Simon (ρ) distribution is
where is the gamma function. Thus, if is an integer,
The parameter can be estimated using a fixed point algorithm.
The probability mass function f has the property that for sufficiently large k we have
This means that the tail of the Yule–Simon distribution is a realization of Zipf's law: can be used to model, for example, the relative frequency of the th most frequent word in a large collection of text, which according to Zipf's law is inversely proportional to a (typically small) power of .
The Yule–Simon distribution arose originally as the limiting distribution of a particular stochastic process studied by Yule as a model for the distribution of biological taxa and subtaxa. Simon dubbed this process the "Yule process" but it is more commonly known today as a preferential attachment process. The preferential attachment process is an urn process in which balls are added to a growing number of urns, each ball being allocated to an urn with probability linear in the number the urn already contains.
The distribution also arises as a compound distribution, in which the parameter of a geometric distribution is treated as a function of random variable having an exponential distribution. Specifically, assume that follows an exponential distribution with scale or rate :
Then a Yule–Simon distributed variable K has the following geometric distribution conditional on W:
The pmf of a geometric distribution is
for . The Yule–Simon pmf is then the following exponential-geometric compound distribution:
The maximum likelihood estimator for the parameter given the observations is the solution to the fixed point equation
where are the rate and shape parameters of the gamma distribution prior on .
generalize the algorithm to Bayesian settings with the compound geometric formulation described above. Additionally, Roberts and Roberts  are able to use the Expectation Maximisation (EM) framework to show convergence of the fixed point algorithm. Moreover, Roberts and Roberts  derive the sub-linearity of the convergence rate for the fixed point algorithm. Additionally, they use the EM formulation to give 2 alternate derivations of the standard error of the estimator from the fixed point equation. The variance of the estimator is
the standard error is the square root of the quantity of this estimate divided by N.
The two-parameter generalization of the original Yule distribution replaces the beta function with an incomplete beta function. The probability mass function of the generalized Yule–Simon(ρ, α) distribution is defined as
with . For the ordinary Yule–Simon(ρ) distribution is obtained as a special case. The use of the incomplete beta function has the effect of introducing an exponential cutoff in the upper tail.
Computer software for the Yule-Simon distribution
Yule-Simon distribution using Python
The Yule-Simon distribution is available in the python scipy library. 
- Colin Rose and Murray D. Smith, Mathematical Statistics with Mathematica. New York: Springer, 2002, ISBN 0-387-95234-9. (See page 107, where it is called the "Yule distribution".)
- Simon, H. A. (1955). "On a class of skew distribution functions". Biometrika. 42 (3–4): 425&ndash, 440. doi:10.1093/biomet/42.3-4.425.
- Garcia Garcia, Juan Manuel (2011). "A fixed-point algorithm to estimate the Yule-Simon distribution parameter". Applied Mathematics and Computation. 217 (21): 8560–8566. doi:10.1016/j.amc.2011.03.092.
- Yule, G. U. (1925). "A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S". Philosophical Transactions of the Royal Society B. 213 (402–410): 21&ndash, 87. doi:10.1098/rstb.1925.0002.
- Roberts, Lucas; Roberts, Denisa (2017). "An Expectation Maximization Framework for Preferential Attachment Models". arXiv:1710.08511 [stat.CO].