Conditional Expectation

A Functions of Random Variables $h(x)=E[Y|X=x]$.

Definition

With Random Variable $X,Y$ The conditional expectation of $Y$ given the value of $X$ is:

$$E[Y|X=x]=\{\begin{array}{l}{\sum }_{y}y{p}_{Y|X}(y|x)\\ \\ {\int }_{-\infty }^{\infty }y{f}_{Y|X}(y|x)dy\end{array}$$

Intuition

The conditional expectation of $Y|X$ is the linear function of $X$ describing mean level of $Y$ based on $X$

Properties

• $Y\perp X⟹E(Y|X)=E(Y)$
• $E[g(X)\cdot Y|X]=g(X)E(Y|X)$
• $E[aY+bZ|X]=aE(Y|X)+bE(Z|X)$
• If $Y\perp X⟹E[Y\cdot Z|X]=E(Y|X)\times E(Z|X)$