In probability and statistics, the Yule–Simon distribution is a discrete probability distribution named after Udny Yule and Herbert A. Simon. Simon originally called it the Yule distribution. The probability mass function (pmf) of the Yule–Simon (ρ) distribution is for integer and real , where is the beta function. Equivalently the pmf can be written in terms of the rising factorial as 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 . Occurrence The Yule–Simon distribution arose originally as the limiting distribution of a particular model studied by Udny Yule in 1925 to analyze the growth in the number of species per genus in some higher taxon of biotic organisms. The Yule model makes use of two related Yule processes, where a Yule process is defined as a continuous time birth process which starts with one or more individuals. Yule proved that when time goes to infinity, the limit distribution of the number of species in a genus selected uniformly at random has a specific form and exhibits a power-law behavior in its tail. Thirty years later, the Nobel laureate Herbert A. Simon proposed a time-discrete preferential attachment model to describe the appearance of new words in a large piece of a text. Interestingly enough, the limit distribution of the number of occurrences of each word, when the number of words diverges, coincides with that of the number of species belonging to the randomly chosen genus in the Yule model, for a specific choice of the parameters. This fact explains the designation Yule–Simon distribution that is commonly assigned to that limit distribution. In the context of random graphs, the Barabási–Albert model also exhibits an asymptotic degree distribution that equals the Yule–Simon distribution in correspondence of a specific choice of the parameters and still presents power-law characteristics for more general choices of the parameters. The same happens also for other preferential attachment random graph models. The preferential attachment process can also be studied as an