I am preparing an exam and I have found in my lecture notes the two following formulas that the professor uses again and again, but I have no clue where they come from:
$\mathbb{E}(X|Y,Z)=\mathbb{E}(X|Z)+\mathbb{C}\mathrm{ov} (X,Y|Z)\cdot [\mathbb{V}\mathrm{ar} (Y|Z)]^{-1}\cdot[Y-\mathbb{E}(Y|Z)]$
and
$\mathbb{V}\mathrm{ar} (X|Y,Z)=\mathbb{V}\mathrm{ar} (X|Z)-\mathbb{C}\mathrm{ov} (X,Y|Z)\cdot [\mathbb{V}\mathrm{ar} (Y|Z)]^{-1}\cdot\mathbb{C}\mathrm{ov} (X,Y|Z)$.
Since we assume to know the distributions of $(X,Y|Z)$ and $(Y|Z)$, I thought to express the distribution of $(X|Y,Z)=\dfrac{(X,Y,Z)}{(Y,Z)}=\dfrac{(X,Y|Z)\cdot (Z)}{(Y|Z)\cdot (Z)}= \dfrac{(X,Y|Z)}{(Y|Z)}$ and then apply an approximation of order 2 for the expectation and variance of a ratio (like here), but I'm far from obtaining the two formulas written above...
Could any of you please give me a hint?
Many thanks. Sabina