Using principal axis theorem prove that density function of multivariate normal distribution can be represented as multiplication of densities of univariate normal distributions.
I've been given only very basic definitions of probability theory and I struggle a lot with proving this statement.
I have $$p(x) = \frac1{\sqrt{2\pi\sigma^2}}e^{-\frac{(x - a)^2}{2\sigma^2}}$$ what is density function of univariate normal distribution. Next there is $$p(x) = \frac1{(2\pi)^{n/2}|\Sigma|^{1/2}}e^{-\frac12(x - a)^T\Sigma^{-1}(x - a)}, \quad x \subseteq \mathbb{R}^n$$ what is density function of multivariate normal distribution. I think the main trouble for me is understanding how $\sigma$ transform to $\Sigma$ and I don't know when should I apply principal axis theorem.