# Conditional Expectation and Variance

Properties:

### Conditional Expectation

$$E(X|Y)$$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: $$E(g(S)T | S) = g(S)E(T|S)$$

### Conditional Variance

If $$Var(Y)$$is difficult to find directly, we can use the **variance decomposition** to condition the variance on another variable.

$$
Var(Y) = E(Var(Y|X)) + Var(E(Y|X))
$$