# combining variables from different tables

I'm having trouble combining variables from differents tables.

In this case, I have the exogenous variables $$X$$ (categorical) and the endogenous variable $$Y$$ (continuous). None of these variables are independant.

• Table A has the variables $$x_1, x_2, x_3$$ and $$Y$$.
• Table B has the variables $$x_1, x_4$$ and $$Y$$.
• Table C has the variables $$x_1, x_2, x_3, x_4$$, whithout any reference to $$Y$$.

It's easy to compute from the table A $$E [ Y | x_1, x_2, x_3 ]$$, and from table B $$E [ Y | x_1, x_4 ]$$. But my main goal is to compute $$E [ Y | x_1, x_2, x_3, x_4 ]$$ and I can't find a way to do it.

It might help to know that from table C, I can also get the conditional probability $$P[x_4 | x_1, x_2, x_3]$$ as well as $$P[x_2 | x1]$$ and $$P[x_3 | x1]$$.

One of my try was to use the formula $$E[Y] = E[E[Y|X]]$$ (with all $$x_i$$) as I am able to know $$E[Y]$$. But then I would have to solve the following sum $$\sum y \cdot P[Y | x_1, x_2, x_3, x_4]$$, and I believe I can't build the requiered values for $$y$$ (or don't know how).

Any idea how to get $$E [ Y | x_1, x_2, x_3, x_4 ]$$ ?