Suppose I have a random vector $x=[x_1,...,x_k]$ s.t. $x∼N(\mu,\sum)$. How is the length or magnitude of $x$ distributed?
I know that if $k=2$ and $\sigma_1=\sigma_2$ and $\sigma_{12}=0$ ($x_1$ and $x_2$ are not correlated), it is Rayleigh distribution.
I also know that $\sqrt{\sum_{i=1}^k(\frac{x_i-\mu_i}{\sigma_i})^2}$ is Chi distributed (no correlation). However, the random variables are normalized by its standard deviation, it is just the length of a zero-mean unit variance Gaussian vector.
If it is not zero mean, we can have noncentral chi distribution. It is non-zero-mean but still unit variance Gaussian vector.
So my question is:
When $\sigma_i$ has different values for all $i=1,...,k$, what is the distribution of vector length/magnitude $|x|=\sqrt{\sum_{i=1}^k x_i^2}$?
When the random variables are correlated, what is the distribution of the vector length/magnitude $|x|=\sqrt{\sum_{i=1}^k x_i^2}$?