Conditional expectation multiple variables I was wondering what the definition of conditional expectation on more than one variables is.
Is $E[X|Y_{1},Y_{2}]=\int x f_{x|y1,y2}dx$?
 A: The general definition says that in the second component of the conditional expectation is a $\sigma$-algebra. And $E(X|Y)$ is an abbreviation for $E(X|\sigma(Y))$ where $\sigma(Y)$ is the induced $\sigma$-algebra of $Y$.
Therefore $$E(X|Y_1,Y_2):=E(X|\sigma(Y_1,Y_2)).$$
Further the general definition for this case: The conditional expectation of $X$ given $\sigma(Y_1,Y_2)$ is a random variable $E(X|\sigma(Y_1,Y_2))=Z$ such that 


*

*$Z$ is $(\sigma(Y_1,Y_2), \mathcal{B}({\mathbb{R}}))$-measurable

*$\int_A X ~\text{d}P = \int_A Z ~\text{d}P$ for all $A\in \sigma(Y_1,Y_2)$


where we call $Z$ a version of $E(X|\sigma(Y_1,Y_2)).$
You should work with this. Or is your aim to explicitly write it as an integral?
Then I'd say $$g(y_1,y_2)=\int x f_{X|Y_1=y_1,Y_2=y_2}(x) dx$$ defines a version $g(Y_1,Y_2)$ of $E(X|Y_1,X_2)$ where
$$f_{X|Y_1=y_1,Y_2=y_2}(x):=\begin{cases} \frac{f_{(X,Y_1,Y_2)}(x,y_1,y_2)}{f_{(Y_1,Y_2)} (y_1,y_2)} &\text{ if } f_{(Y_1,Y_2)}(y_1,y_2)>0 \\ 0 &\text{ else } \end{cases} $$
where $(x,y_1,y_2) \mapsto f_{(X,Y_1,Y_2)}(x,y_1,y_2)$ is the joint density and the marginal density is defined as 
$$f_{(Y_1,Y_2)}(y_1,y_2):=\int f_{(X,Y_1,Y_2)} (x,y_1,y_2) dx.$$
