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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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Wilks,   "Mathematical statistics" , Wiley  (1962)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T.S. Ferguson,   "A Bayesian analysis of some nonparametric problems"  ''Ann. Stat.'' , '''1'''  (1973)  pp. 209–230</TD></TR></table>