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Mean & Co-variance of random vectors
matrix with element equal to
Suppose A = p x m matrix, b, C = q x n matrix, d .
Proof: ith element ofso
ith element of
Proof involves first considering theelement of and then taking the covariance of both of these elements. By properties of 1D covariances, it works out that the element is just
And obviouslyis symmetric thus by definition is pd.
Useful results of multivariate normals
is multivariate normal if can be written as:
for some non-random matrix and non-random vector with with iid N(0,1).
Defining, we have and