I have fitted a maximum likelihood Gaussian distribution $N(\mu, \Sigma)$ on a multidimensional data set $X$. I wonder how would $p(X)$ change if one dimension of $X$ is scaled by a factor?
It's clear how $\mu$ and $\Sigma$ changes, but I couldn't see how it affects the exponent correspondingly $$(x-\mu)'\Sigma^{-1}(x-\mu)$$ Any help would be appreciated, thanks!
Suppose your original vector $X = [X_1,\ldots,X_n]^T$ and without loss of any generality, $X'_n = \rho X_n$ where $\rho$ is a scalar, then I can rewrite the new observation vector as $X' = A X$, where \begin{align} A = \begin{bmatrix} 1 & & & \\ & \ddots & & \\ & & 1 & \\ & & & \rho \end{bmatrix}. \end{align} Therefore the distribution of $X' \sim \mathcal{N}(A \mu, A \Sigma A^T)$. Now, let us write the exponent term explicitly \begin{align} (X' - A \mu)^T (A \Sigma A^T)^{-1} (X' - A \mu) &= (X - \mu)^T A^T (A \Sigma A^T)^{-1} A (X - \mu) \\ &= (X - \mu)^T A^T A^{-T} \Sigma^{-1} A^{-1} A (X-\mu) \\ &= (X - \mu)^T \Sigma^{-1} (X -\mu) \end{align}
Note that as $A$ is a diagonal matrix, it is invertible.
