Law of iterated expectations

$E(y|\mathbf{x})=E[E(y|\mathbf{w})|\mathbf{x}]$

• $\mathbf{w}$ is a random vector
• $y$ is a random variable
• $\mathbf{x}$ is a random vector and functionally derived $\mathbf{x}=f(\mathbf{w})$ => This statement implies that if we know the outcome of $\mathbf{w}$, then we know the outcome of $\mathbf{x}$ => Eselsbrücke: “The smaller information set always dominates”

$E[X]=E[E[X|Y]]$

In plain English, the expected value of X is equal to the expectation over the conditional expectation of X given Y. More simply, the mean of X is equal to a weighted mean of conditional means.