Conditional expectation on multiple variables

[dabd]

How to compute E[X|Y1,Y2]?
Assume all random variables are discrete.

I tried E[X|Y1,Y2] = \sum_x{x p(x|y1,y2) but I'm not sure how to compute p(x|y1,y2] = \frac{p(x \cap y1 \cap y2)}{p(y1 \cap y2)}

If it is correct, how can I simplify the expression if Y1 and Y2 are iid?

 

[EnumaElish]

If y1 and y2 are independent then p(y1, y2) = p(y1)p(y2).

 

[statdad]

In general

<br /> p(x \mid y_1, y_2) = \frac{p(x,y_1,y_2)}{p(y_1,y_2)}<br />

where the numerator is the joint density (or mass function for discrete case) of all three, and the denominator is the marginal of the two ys. You treat this as a function of x alone. Then, in the discrete case, the expected value is

<br /> \sum x p(x \mid y_1, y_2)<br />

and in the continuous case it is

<br /> \int x p(x \mid y_1, y_2) \, dx<br />

In each case it is possible for the answer to depend on both y_1, y_2.