2
$\begingroup$

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

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

How can we derive the closed form expression for the following PDF?

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

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

  • 1
    Duplicate of http://stats.stackexchange.com/q/27413/66332012-04-30
  • 0
    The closed form is quite unlikely. One is first to consider $X|(Y,Z)$, then represent $(Y,Z) = A (\cos(\Phi), \sin(\Phi))$ and then average out $\Phi$. I would expect appearance of some [matrix Bessel functions](http://www.jstor.org/stable/1969810). Has this example its origin in wireless communication theory, by chance?2012-04-30
  • 0
    Thanks for the hints. Actually, the question is originated from speech enhancement domain.2012-04-30
  • 0
    smo: Any luck with the answer below?2012-05-07

1 Answers 1