Poisson-Dirichlet distribution
In probability theory, a branch of mathematics Poisson-Dirichlet distributions are probability distributions on the set of nonnegative, non-decreasing sequences with sum 1, depending on two parameters  and . It can be defined as follows. One considers independent random variables   such that  follows the beta distribution of parameters  and . Then, the Poisson-Dirichlet distribution  of parameters  and  is the law of the random decreasing sequence containing  and the products . This definition is due to Jim Pitman and Marc Yor. It generalizes Kingman's law, which corresponds to the particular case . Number theory  
Patrick Billingsley has proven the following result:  if  is a uniform random integer in , if  is a fixed integer, and if  are the  largest prime divisors of  (with  arbitrarily defined if  has less than  prime factors), then the joint distribution ofconverges to the law of the  first elements of a  distributed random sequence, when  goes to infinity. Random permutations and Ewens's sampling formula  
The Poisson-Dirichlet distribution of parameters  and  is also the limiting distribution, for  going to infinity, of the sequence , where  is the length of the  largest cycle of a uniformly distributed permutation of order . If for , one replaces the uniform distribution by the distribution  on  such that , where  is the number of cycles of the permutation , then we get the Poisson-Dirichlet distribution of parameters  and . The probability distribution  is called Ewens's distribution, and  comes from the Ewens's sampling formula, first introduced by Warren Ewens in population genetics, in order to describe the probabilities associated with counts of how many different alleles are observed a given number of times in the sample. References 

Probability distributions