Proof that multivariate normal has mean mu

Hi all,

Does anyone know how to show that the expectation of a multivariate normal random variable (X) is mu, by integrating X*pdf? It looks like the determinant of the variance-covariance matrix should come out somehow, but I guess my matrix calculus skills are too weak to figure out how this happens.

Any help would be greatly appreciated.

Thanks!

Re: Proof that multivariate normal has mean mu

Hey bsk.

Do you have any vector identities of derivatives and integrals respect to taking the derivatives of matrix objects that have variables in them?

Re: Proof that multivariate normal has mean mu

Hi Chiro,

Thanks for the reply. I'm afriad I don't quite understand your question though.

Re: Proof that multivariate normal has mean mu

Its basically relationships by doing differentiation and integration with vectors and matrices instead of real numbers.

So things like d/dx (Ax) where A is a matrix and x is a vector for example each element of x is independent to the others and you get a result in terms of x. Also things like d/dx (x^t*A*x). These results do exist because they are used for derivations of least squares for multiple linear regression.