(Solution) - Prove Theorem 4 7 2 for the case in which X and -(2025 Original AI-Free Solution)

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Prove Theorem 4.7.2 for the case in which X and Y have a discrete joint distribution. The key to the proof is to write all of the necessary conditional p.f.?s in terms of the joint p.f. of X and Y and the marginal p.f. of X. To facilitate this, for each x and z, give a name to the set of y values such that r(x, y) = z.