There are frequently situations where you have information on the conditional distribution of Y given X, but are interested in the conditional distribution of X given Y
Recalling Pr(Y = y = x) = , derive a relationship between Pr(X = x = y) and Pr(Y = y = x). This is called Bayes' theorem.
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Answer:
Given Pr(Y = y = x) = ,
Pr(Y = y = x) × Pr(X = x) = Pr(X = x, Y = y);
similarly Pr(X = x = y) = and
Pr(X = x = y) × Pr(Y = y) = Pr(X = x, Y = y). Equating the two and solving for Pr(X = x = y) then results in
Pr(X = x = y) = .
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