# Conditional Expectation and Variance

Properties:

$E(X|Y)$

is the conditional expectation of $X$

given $Y$

- $E(X|Y=y)$is a fixed value, but$E(X|Y)$is a random variable (it is a function of$Y$)
- Iterated expectation:$E(E(X|Y)) = E(X)$
- Additivity:$E(Y+Z | X) = E(Y|X) + E(Z|X)$
**does not work**on the right hand side:$E(Y | X+Z) \ne E(Y|X) + E(Y|Z)$

- Linearity:$E(aX + b | Y) = aE(X|Y) + b$
- Conditioning on the same variable:$E(g(S)T | S) = g(S)E(T|S)$

If

$Var(Y)$

is difficult to find directly, we can use the**variance decomposition**to condition the variance on another variable.$Var(Y) = E(Var(Y|X)) + Var(E(Y|X))$

Last modified 2yr ago