Law of total expectation

The proposition in probability theory known as the law of total expectation,[1] the law of iterated expectations[2] (LIE), Adam's law,[3] the tower rule,[4] and the smoothing theorem,[5] among other names, states that if X {\displaystyle X} is a random variable whose expected value E ⁡ ( X ) {\displaystyle \operatorname {E} (X)} is defined, and Y {\displaystyle Y} is any random variable on the same probability space, then