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I am interested in finding their joint density function P(A and B) and associated properties such as maximas, minimas, etc.

Ideally the joint density is expressed as a closed form 2D mixture model as well, but this is not critical.

I could do something perhaps by brute force by use of Baye's theorem:

ie. I can approximate

P(A and B) = P(A) P(B | A) = P(B) | P(A | B)

But eventually I need to extend this to higher dimensions, eg. P( A and B and C and D... etc) and this is certainly no trivial task.