# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 38: Orthogonality Property of Conditional Expectation

The orthogonality property of conditional expectation states that, where $X, Y$ are random variables, $\textrm{E}[Y|X]$ is the conditional expectation of $Y$ dependent on $X$, and $\phi (X)$ is any function of $X$,

$\textrm{E}(Y - \textrm{E}[Y|X]) \phi(X)) = 0$.

This can be interpreted as meaning that the ‘vector’/function $Y - \textrm{E}[Y|X]$ is orthogonal to the space of functions of $X$, which implies that $\textrm{E}[Y|X]$ gives the minimum mean squared error of $Y$ given $X$.