Dirichlet distribution

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A probability distribution on the simplex

where determined by the probability density

where and

where is the gamma-function. If , one has a special case of the Dirichlet distribution: the beta-distribution. The Dirichlet distribution plays an important role in the theory of order statistics. For instance, if are independent random variables that are uniformly distributed over the interval and are the corresponding order statistics (cf. Order statistic), the joint distribution of the differences

(it is assumed that ) has the Dirichlet distribution with , , .


[1] S.S. Wilks, "Mathematical statistics" , Wiley (1962)



[a1] T.S. Ferguson, "A Bayesian analysis of some nonparametric problems" Ann. Stat. , 1 (1973) pp. 209–230
How to Cite This Entry:
Dirichlet distribution. L.N. Bol'shev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098