Artificial Intelligence - Probability and Bayesian Networks

Problem 3 (10 points.) Prove that the two definitions of conditional independence of random variables are equivalent. Let X, Y, Z be random variables. The two definitions are: Definition 1: X and Y are conditionally independent given Z if for any value z of X, any value y of Y, and any value z of Z, the following holds: px, ) p2) x p(2). Definition 2: X and Y are conditionally independent given Z if for any value of X, any value y of Y, and any value z of Z, the following holds: p(xly, 2) - p(x|2)