The proposition in probability theory known as the law of total expectation, the law of iterated expectations, Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X {\displaystyle X} is a random variable whose expected value E {\displaystyle \operatorname {E} } is defined, and Y {\displaystyle Y} is any random variable on the same probability space, then E = E , {\displaystyle \operatorname {E} =\operatorname {E},} i.e., the expected value of the conditional expected value of X {\displaystyle X} given Y {\displaystyle Y} is the same as the expected value of X {\displaystyle X}.
