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The joint distribution of the elements from the sample covariance matrix of observations from a multivariate [[Normal distribution|normal distribution]]. Let the results of observations have a  $  p $-dimensional normal distribution  $  N( \mu , \Sigma ) $
with vector mean  $  \mu $
and covariance matrix  $  \Sigma $. 
Then the joint density of the elements of the matrix  $  A= \sum _ {i= 1}  ^ {n} ( X _ {i} - \overline{X}\; ) ( X _ {i} - \overline{X}\; )  ^  \prime  $
is given by the formula

$$ 
w( n, \Sigma )  =  
\frac{| A | ^ {( n- p)/2 }
e ^ {-  \mathop{\rm tr}  ( A \Sigma  ^ {- 1} )/2 } }{2 ^ {( n- 1) p/2 } \pi ^ {p( p- 1)/4 } | \Sigma | ^ {( n- 1)/2 }
\prod _ { i= 1} ^ { p }  \Gamma \left ( n- 
\frac{i}{2}
 \right ) }

$$

( $   \mathop{\rm tr}  M $
denotes the trace of a matrix  $  M $), 
if the matrix  $  \Sigma $
is positive definite, and  $  w( n, \Sigma )= 0 $
in other cases. The Wishart distribution with  $  n $
degrees of freedom and with matrix  $  \Sigma $
is defined as the  $  p( n+ 1)/2 $-
dimensional distribution  $  W( n, \Sigma ) $
with density  $  w( n, \Sigma ) $. 
The sample covariance matrix  $  S= A/( n- 1) $, 
which is an estimator for the matrix  $  \Sigma $, 
has a Wishart distribution.

The Wishart distribution is a basic distribution in multivariate statistical analysis; it is the  $  p $-dimensional generalization (in the sense above) of the  $  1 $-dimensional [[Chi-squared distribution| "chi-squared"  distribution]].

If the independent random vectors  $  X $
and  $  Y $
have Wishart distributions  $  W( n _ {1} , \Sigma ) $
and  $  W ( n _ {2} , \Sigma ) $, 
respectively, then the vector  $  X + Y $
has the Wishart distribution  $  W( n _ {1} + n _ {2} , \Sigma ) $.

The Wishart distribution was first used by J. Wishart [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Wishart,   ''Biometrika A'' , '''20'''  (1928)  pp. 32–52</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Anderson,   "An introduction to multivariate statistical analysis" , Wiley  (1958)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Khirsagar,   "Multivariate analysis" , M. Dekker  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.J. Muirhead,   "Aspects of multivariate statistical theory" , Wiley  (1982)</TD></TR></table>