5. (15 pts) Let's look at the Bayesian networks below. We use the notation X _ Y to denote that the variable X is independent of Y, and (X1 IL X2|Y) to denote X] is independent of X2 conditioned on Y. B A E B A M M A (i) (ii) (iii (a) (5 pts) For each of these three network, write the factored joint distribution. For example, for a BN with structure X - Y, the factored joint distribution is P(X, Y) = P(X)P(Y(X). (b) (2 pts) Using the joint distribution you wrote down for Fig(i), write down a formula for P( B, E). (c) (2 pts) Now prove that B I E in Fig(i). (d) (6 pts) Similarly, prove that B IL MA in the Bayesian network of Fig(ii), and M II JJA in the Bayesian network of Fig(iii)