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).