1. Suppose X is a random variable and A1. . ... A, are disjoint events that form a partition of the sample space. For any event B with P(A; ( B) > 0 for all i, prove the following (a) For event C, we have P(C B) = J P(A B)P(CIA, n B). i=1 (b) Use part (a) to show that PXB(x) = ) P(AB) px Ana(I). i=1 (c) Furthermore, show that E[X B] =) P(A B)EX|AnB). i=1