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??
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1$\begingroup$ @Gue : I know how to find both mean and co variance of the new distribution, but how can I show that new distribution will also be gaussian $\endgroup$– Sahil ChadhaCommented Jun 28, 2019 at 9:17
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1$\begingroup$ An easy way to prove this is by using characteristic functions, see e.g. math.stackexchange.com/questions/605816/… $\endgroup$– sp59b2Commented Jun 28, 2019 at 9:57
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$\begingroup$ There are myriad ways to establish this: see our list of characterizations of Gaussian distributions for some ideas. $\endgroup$– whuber ♦Commented Jun 28, 2019 at 14:10
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1 Answer
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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: $$f_\mathbf{Y}(\mathbf{y})=\frac{1}{\vert\mathbf{A}\vert}f_\mathbf{X}(\mathbf{A}^{-1}\vert\mathbf{y}-\mathbf{b}\vert)$$
Since $\vert\mathbf{A}\vert$ is constant, $f_{\mathbf{Y}}(y)\propto f_\mathbf{X}(\mathbf{A}^{-1}\vert\mathbf{y}-\mathbf{b}\vert)$, and since $f_\mathbf{X}(\mathbf{.})$ is in MV normal form, so is $f_\mathbf{Y}(\mathbf{y})$.
