Evaluation of the expected value of a determinant (Q1120932)

Let \(A=(a_{ij})\), \(i=1,...,k\), \(j=1,...,n\), \(k\leq n\) be a matrix of kn independent random variables with \(\Pr \{a_{ij}=1\}=1/2\) and \(\Pr \{a_{ij}=0\}=1/2\). Then A can be considered as a random drawing of the set of matrices with 1-0 entries. The following theorem is proven: Let \(a_{ij}\), \(i=1,...,k\), \(j=1,...,n\), be uncorrelated random variables with mean \(\mu\) and variance \(\sigma^ 2\). Then \[ {\mathcal E}| AA'| =(n!/(n-k)!)(\sigma^ 2)^{k-1}(\sigma^ 2+k\mu^ 2). \]