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Let $\mathbf{X} \sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$ be a multivariate Normal variable with $d$ dimensions. I'm interested in the marginal distribution of $\|\mathbf{X}\|^2 = \sum_{i=1}^d X_i^2$.

It is well known that if $\mathbf{\mu} = \mathbf{0}$ and $\mathbf{\Sigma} = \mathbf{I}_{d \times d}$, then $\|\mathbf{X}\|^2 \sim \chi^2_{d}$. However, is anything known about the general case, i.e. when the components are correlated?

macjan
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