# Dirichlet distribution

A probability distribution on the simplex

$$S _ {k} = \{ {( x _ {1} \dots x _ {k} ) } : {x _ {1} \geq 0, \ldots, x _ {k} \geq 0 , x _ {1} + \dots + x _ {k} = 1 } \} ,$$

where $k= 2, 3 \dots$ determined by the probability density

$$p ( x _ {1} \dots x _ {k} ) = \left \{ \begin{array}{ll} C _ {k} \prod _ { i=1 } ^ { k } x _ {i} ^ {\nu _ {i} - 1 } & \textrm{ if } ( x _ {1}, \ldots, x _ {k} ) \in S _ {k} , \\ 0 & \textrm{ if } ( x _ {1}, \ldots, x _ {k} ) \notin S _ {k} , \\ \end{array} \right .$$

where $\nu _ {1} > 0 \dots \nu _ {k} > 0$ and

$$C _ {k} = \Gamma ( \nu _ {1} + \dots + \nu _ {k} ) \prod _ { i=1 } ^ { k } \frac{1}{\Gamma ( \nu _ {i} ) } ,$$

where $\Gamma ( \cdot )$ is the gamma-function. If $k= 2$, 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 $X _ {1}, \ldots, X _ {n}$ are independent random variables that are uniformly distributed over the interval $[ 0, 1]$ and $X ^ {( 1)} \leq \dots \leq X ^ {( n)}$ are the corresponding order statistics (cf. Order statistic), the joint distribution of the $k$ differences

$$X ^ {( m _ {1} ) } , X ^ {( m _ {2} ) } - X ^ {( m _ {1} ) }, \ldots, X ^ {( m _ {k-1} ) } - X ^ {( m _ {k-2} ) } , 1 - X ^ {( m _ {k} ) }$$

(it is assumed that $1 \leq m _ {1} < m _ {2} < \dots < m _ {k-1}$) has the Dirichlet distribution with $\nu _ {1} = m _ {1}$, $\nu _ {2} = m _ {2} - m _ {1}, \ldots, \nu _ {k-1} = m _ {k-1} - m _ {k-2}$, $\nu _ {k} = n - m _ {k-1}$.

#### References

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

#### References

 [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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_distribution&oldid=51247
This article was adapted from an original article by L.N. Bol'shev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article