## Deriving Conditional Probability Distribution

Given the following joint normal PDF of $X \in \mathcal{R}^K$ and $Y,Z \in \mathcal{R}$

$p(\begin{bmatrix} X \\ Y \\ Z \end{bmatrix}) = \mathcal{N}(\begin{bmatrix} \mu_X \\ \mu_Y \\ \mu_Z \end{bmatrix},\begin{bmatrix} \Sigma_{XX} \ \Sigma_{XY} \ \Sigma_{XZ} \\ \Sigma_{YX} \ \Sigma_{YY} \ \Sigma_{YZ} \\ \Sigma_{ZX} \ \Sigma_{ZY} \ \Sigma_{ZZ} \end{bmatrix})$

Can we derive the closed form expression for the following PDF?

$P(X|A)$ (or equivalently $P(X|A^2)$)

where, $A = \sqrt{Y^2+Z^2}$

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