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# How to add two normal distributions?

Posted on 2007-03-29

Is there an efficient way to add two normal distributions?

Let's say I have two multivariate normal distributions with means m1 and m2, and covariance matrices C1 and C2, and that the number of elements in each distribution is n1 and n2.

The mean of the sum of the distributions would then be

(m1 * n1 + m2 * n2) / (n1 + n2)

But is there an efficient way to calulate the new covariance matrix, other than iterating over all of the points of the two distributions? I have a feeling there should be, but I can't see it now.