User talk:Chundt

Hi Michelle!

Here is a helpful formula:
 * $$\mathrm{Var}(X) = \mathrm{E}[X^2] - (\mathrm{E}[X])^2\,$$

See if you can prove it!

Here is the proof that $$\mathrm{E}[\mathrm{E}[X|Y]] = \mathrm{E}[X]\,$$ for any $$X\,$$:

Let $$\mathbb{X}$$ be the range of $$X\,$$ and $$\mathbb{Y}$$ the range of $$Y\,$$. Then



\mathrm{E}[\mathrm{E}[X|Y]] = \sum_{y \in \mathbb{Y}}p_Y(y)\mathrm{E}[X|Y=y]$$
 * $$ = \sum_{y \in \mathbb{Y}}p_Y(y)\sum_{x \in \mathbb{X}}xp_{X|Y}(x|y)$$
 * $$ = \sum_{y \in \mathbb{Y}}p_Y(y)\sum_{x \in \mathbb{X}}x\frac{p_{X,Y}(x,y)}{p_Y(y)}$$
 * $$ = \sum_{y \in \mathbb{Y}}\sum_{x \in \mathbb{X}}xp_Y(y)\frac{p_{X,Y}(x,y)}{p_Y(y)}$$
 * $$ = \sum_{y \in \mathbb{Y}}\sum_{x \in \mathbb{X}}xp_{X,Y}(x,y)$$
 * $$ = \sum_{x \in \mathbb{X}}\sum_{y \in \mathbb{Y}}xp_{X,Y}(x,y)$$
 * $$ = \sum_{x \in \mathbb{X}}x\sum_{y \in \mathbb{Y}}p_{X,Y}(x,y)$$
 * $$ = \sum_{x \in \mathbb{X}}xp_X(x)$$
 * $$ = \mathrm{E}[X]\,$$