law of iterated expectation law of total expectation #probability Proposition E[E[Y|X]]=E[Y]E\left\lbrack E\left\lbrack Y|X \right\rbrack \right\rbrack = E\lbrack Y\rbrack E[E[g(X,Y) | X]]=E[g(X,Y)]E\left\lbrack E\left\lbrack g(X,Y)~|~X \right\rbrack \right\rbrack = E\left\lbrack g(X,Y) \right\rbrack See also expectation conditional probability