Let (Z_{1},...,Z_{n}) be a sequence of n independent standard Gaussian random variables. Consider two non-central chi squares random variables X=\sum_{i=1}^n (Z_{i}-a_{i})^{2}and Y=\sum_{i=1}^n (Z_{i}-b_{i})^{2}, where (a_{1},...,a_{n}) and (b_{1},...,b_{n}) 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