CHIC 465/565 – Environmental Epidemiology 8 Concluding Remarks Bibliography

Appendix A Miscellaneous Results

Theorem A.1.

Let $X$ and $Y$ be random variables with probability densities $\pi(x)$ and $\pi(y)$ respectively. The Tower Law for expectations states:

\mathbb{E}(X)=\mathbb{E}[\mathbb{E}(X|Y)].

This the most common way that the theorem is stated, however, to make things more clear, we can make explicit the distributions over which the expectations are taken:

\mathbb{E}_{X}(X)=\mathbb{E}_{Y}[\mathbb{E}_{X|Y}(X|Y)]

Proof:

The proof of this theorem is straightforward and involves expanding the definition of the expectation:

$\displaystyle\mathbb{E}_{X}(X)$	$\displaystyle=$	$\displaystyle\int x\pi(x)\mathrm{d}x$
	$\displaystyle=$	$\displaystyle\int x\left\{\int\pi(x,y)\mathrm{d}y\right\}\mathrm{d}x$
	$\displaystyle=$	$\displaystyle\int\left\{\int x\pi(x\|y)\pi(y)\mathrm{d}y\right\}\mathrm{d}x$
	$\displaystyle=$	$\displaystyle\int\left\{\int x\pi(x\|y)\pi(y)\mathrm{d}x\right\}\mathrm{d}y$
	$\displaystyle=$	$\displaystyle\int\pi(y)\left\{\int x\pi(x\|y)\mathrm{d}x\right\}\mathrm{d}y$

Hence

\mathbb{E}_{X}(X)=\mathbb{E}_{Y}[\mathbb{E}_{X|Y}(X|Y)].

$\Box$