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Given a multivariate normal distribution

$f(x) = \frac{1}{\sqrt{(2 \pi)^n|\Sigma|}} \times \exp\left( -\frac{1}{2} (x-\mu)' \Sigma^{-1} (x-\mu)\right)$

how can I prove that $ (x-\mu)' \Sigma^{-1} (x-\mu) \sim \chi^2_{p} $?

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Note that a random vector $X\sim N(\mu,\Sigma)$ has the pdf in your post when $\Sigma$ is assumed to be positive definite. So there exists a nonsingular matrix $P$ such that $\Sigma=PP'$.

Rewrite the quadratic form as

\begin{align} (X-\mu)' \Sigma^{-1}(X-\mu)&=(X-\mu)'(P')^{-1}P^{-1}(X-\mu) \\&=(X-\mu)'(P^{-1})'P^{-1}(X-\mu) \\&=Y'Y \end{align}

, where $Y=P^{-1}(X-\mu)$.

Now if you can show that $Y$ itself has a multivariate normal distribution with mean vector $E(Y)=0$ and dispersion matrix $P^{-1}\Sigma (P^{-1})'=P^{-1}PP' (P')^{-1}=I$, you are done.

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