Let (Z1,...,Zn) be a sequence of n independent standard Gaussian random variables. Consider two non-central chi squares random variables X=\sum_{i=1}^n (Zi-ai)2 and Y=\sum_{i=1}^n (Zi-bi)2, where (a1,...,an) and (b1,...,bn) are two different vectors of known constants.

I'm looking for the joint density of (X,Y). In particular, what is the probability Pr[X < A & Y < B] for two given constants A,B? Any comments, explanations, or references?

Thanks