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Let $\mathbf{X} = [X_0, X_1]^t \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ with $\boldsymbol{\mu} = [\mu_0, \mu_1]^t \in \mathbb{R}^2$ and $ \boldsymbol{\Sigma} = \begin{bmatrix} \sigma_0^2 & \rho\sigma_0\sigma_1 \\[0.3em] \rho\sigma_0\sigma_1 & \sigma_1^2 \\[0.3em] \end{bmatrix}$. Can we derive a probability density function for the random variable $Y = \|\mathbf{X}\| = \sqrt{X_0^2+X_1^2}$, having $\rho \neq 0$ ? I would expect the solution -that is, if it exists- to be valid for higher dimensions.

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