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In the multivariate case it is proven that if

X~Nm(U, S) and Y = a + CX where C is an invertible n x n matrix, and S is the covariance matrix then:

Y ~ Nm(a + CU, CSC')

I am trying to apply this proof to a similar problem.

If we have X1...Xk with Xi ~ Nm(Ui, Si) where i = 1...k, with Xi independent.

what is the distribution of Y = a + sigma (i=1 to k) CiXi

I'm trying to apply the change of variable formula here to derive this, but I don't know how to use it for a function of several variables. In the proof with only X I can understand it.

Here Y = a + (C1X1 + C2X2 + C3X3 + ... + CkXk)