Law of Iterated Expectations

metrics EC484_Econometrics_Analysis

It is the mean of $Y$ is the weighted average of the conditional expectation of $Y$ given $X$, weighted by the prob distribution of $X$. Which is also called as LIE

$$E[E(Y|X)]=E(Y)$$ $$E(Y)=\sum _{i=1}^{l}E(Y|X=x_i)P(X=x_i)$$

The proof of the LIE:

E[E(Y|X)]&=\sum_x E(Y|X=x)P(X=x)\\ &=\sum_x\sum_y yP(Y=y|X=x)P(X=x)\\ &=\sum_x\sum_y yP(Y=y,X=x)\\ &=E(Y) \end{aligned}$$ --- Using Chinese, it is 平均的平均就是总平均。