Law of total expectation

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

${\displaystyle \operatorname {E} (X)=\operatorname {E} (\operatorname {E} (X\mid Y)),}$

i.e., the expected value of the conditional expected value of ${\displaystyle X}$ given ${\displaystyle Y}$ is the same as the expected value of ${\displaystyle X}$.

The conditional expected value ${\displaystyle \operatorname {E} (X\mid Y)}$, with ${\displaystyle Y}$ a random variable, is not a simple number; it is a random variable whose value depends on the value of ${\displaystyle Y}$. That is, the conditional expected value of ${\displaystyle X}$ given the event ${\displaystyle Y=y}$ is a number and it is a function of ${\displaystyle y}$. If we write ${\displaystyle g(y)}$ for the value of ${\displaystyle \operatorname {E} (X\mid Y=y)}$ then the random variable ${\displaystyle \operatorname {E} (X\mid Y)}$ is ${\displaystyle g(Y)}$.

One special case states that if ${\displaystyle {\left\{A_{i}\right\}}}$ is a finite or countable partition of the sample space, then

${\displaystyle \operatorname {E} (X)=\sum _{i}{\operatorname {E} (X\mid A_{i})\operatorname {P} (A_{i})}.}$