An unusual joint probability density function
I have a problem that I'm not at all sure how to approach, and would appreciate any guidance on how to proceed (or if I can proceed at all). I have two discrete random variables, for which I know the probability of all possible (independent) outcomes, but I want to determine the joint pdf and the applicability of these independent probabilities becomes a little questionable.
For example, say:
PX(A) = 0.16
PX(B) = 0.11
PX(C) = 0.07
and also that:
PY(A) = 0.12
PY(B) = 0.17
PY(C) = 0.09
The problem I have is to determine the probability of the outcomes of the joint pdf, given that the combination of these two random variables must agree on the outcome. So, PX,Y(A,B) = PX,Y(A,C) = PX,Y(B,C) = 0, which obviously a little different from the typical outcome here.
I hope that makes some semblance of sense, and that some mathematics that I'm capable of understanding can be brought to bear on the problem (my suspicion is that a fundamental piece of information that would help solve the problem is not available). Any help would be appreciated.