Suppose I have a multivariate Gaussian distribution x and a constant matrix A. I know how to calculate the mean and covariance of Ax but how can I prove that Ax will also be multivariate gaussian??
Contents
hide
Best Answer
In addition to characteristic functions, one can use Jacobians to find an expression for RV transformations, i.e. when we set $mathbf{Y}=mathbf{AX}+mathbf{b}$, it's further simplified to: $$f_mathbf{Y}(mathbf{y})=frac{1}{vertmathbf{A}vert}f_mathbf{X}(mathbf{A}^{-1}vertmathbf{y}-mathbf{b}vert)$$
Since $vertmathbf{A}vert$ is constant, $f_{mathbf{Y}}(y)propto f_mathbf{X}(mathbf{A}^{-1}vertmathbf{y}-mathbf{b}vert)$, and since $f_mathbf{X}(mathbf{.})$ is in MV normal form, so is $f_mathbf{Y}(mathbf{y})$.