Finding the distribution of a function of $n$ random, normally distributed correlated variables Given a random vector $X$ of $n$ normally distributed random variables, and an $n \times n$ covariance matrix of those variables with non-zero correlation terms, what is the generalized methodology to find the distribution of a non-linear function $f(X_1,X_2,\dots,X_n)$ of the random variables of $X$?
That's the general formulation of a problem I'm trying to solve. More specifically, given $6$ normally distributed random variables $x_1 \dots x_6$, what is the probability distribution of $$\sqrt{(x_1 - x_4)^2 + (x_2 - x_5)^2 + (x_3 - x_6)^2}$$ where $x_1,x_2,x_3$ are correlated and $x_4,x_5,x_6$ are correlated (i.e. the upper right and lower left $3 \times 3$ correlation terms are zero, but all other correlation terms are not).
 A: Assuming the $X_i$ have mean 0, $(X_1 - X_4)^2 + (X_2 - X_5)^2 + (X_3 - X_6)^2$ (or any other quadratic form in the $X_i$) has a generalized chi-square distribution.  See 
http://en.wikipedia.org/wiki/Generalized_chi-square_distribution.  If $Y$ is a nonnegative random variable with density $f_Y(y)$, then $S = \sqrt{Y}$ has density 
$f_S(s) = 2 s f_Y(s^2)$.
A: If the covariance matrix of the random column vector $X^T=(X_1,\ldots,X_n)^T$ is a nonsingular matrix $V$, and if they are jointly, not merely separately normally distributed, then $V$ has a positive-definite symmetric square root (found by first doing the spectral decomposition).  Call that $V^{1/2}$.  Then $V^{-1/2}X$ is normally distributed and its entries are not correlated, but in fact are independent, and all the variances are equal to 1.  (If the variables are separately, but not jointly, normally distributed, then this transformation will still make the variances 1 and the covariances 0, but then one might not have independence, and there are other complications.)  This reduces the problem to that of independent standard normals, provided the expected values are 0.
For quadratic forms, we would then have chi-square distributions or (if the means are not all 0) non-central chi-square distributions.
