Finding the distribution of $\|X-\mu \|_\Sigma^2$ with $X \sim N(\mu,\Sigma)$

Let $\Sigma$ be positive definite and symmetric and let $\| x \|_\Sigma := \sqrt{x^t\Sigma^{-1}x}$ be the Mahalanobis norm. Let further $X \sim N(\mu,\Sigma)$ be a $p-$variat normally distributed random vector. Find the distribution of $\| X-\mu \|_\Sigma^2$.

I do not kow how I could do that. Could you help me?

Solution 1:

Observe that \begin{align*} \lVert X-\mu \rVert^2_\Sigma&= (X-\mu)^T \Sigma^{-1} (X-\mu)\\ &= Z^T Z \end{align*} Where $Z=\Sigma^{-\frac{1}{2}} (X-\mu)\sim\mathcal N (0,I)$.

Hence the distribution is a $\chi$-squared distribution with $p$ degrees of freedom.