" The law of total expectation is: E(Y) = E[E(Y|X)]. It can be generalized to the vector case: E(Y) = E[E(Y|X1,X2)]. Further extension: (i) E(Y|X1) = E[E(Y|X1,X2)|X1] (ii) E(Y|X1,X2) = E[E(Y|X1,X2,X3)|X1,X2] " ==================== I understand the law of total expectation itself, but I don't understand the generalizations to the vector case and the extensions. 1) Is E(Y|X1,X2) a random variable? Is E(Y|X1,X2) a function of both X1 and X2? i.e. E(Y|X1,X2)=g(X1,X2) for some function g? 2) Are (i) and (ii) direct consequences of the law of total expectation? (are they related at all?) I don't see how (i) and (ii) can be derived as special cases from it...can somone please show me how? Any help is much appreciated!