The Wick-Isserlis formula expands a joint moment of a centered multivariate normal distribution
into a sum of products of covariances. Let ![C_(ij)=E[X_iX_j]](/images/equations/Wick-IsserlisFormula/Inline1.svg)
, where ![E[Y]](/images/equations/Wick-IsserlisFormula/Inline2.svg)
denotes the expectation
value of 
,
and let 
denote the set partitions of 
into pairs. Then
![E[product_(i=1)^(2m)X_i]=sum_(pi in P_2(2m))product_({i,j} in pi)C_(ij).](/images/equations/Wick-IsserlisFormula/NumberedEquation1.svg) |
An odd joint moment is zero. The number of products in an even moment of order 
is 
, where 
denotes the double factorial
of 
.
For a multivariate normal distribution
with nonzero means, the variables are first centered
and the resulting products are expanded. This formula
is the pairing identity in Isserlis' theorem.
