I think I am approaching this problem incorrectly. I doubt my notation is correct. This is how far I've come in my reasoning and I'm stuck:
Show that:
$E\left( Y| X\right) =E\left( E\left( Y| Z,X\right) | X\right)$
I'm writing the inner expectation as the summation:
$E\left( Y| Z,X\right) =\sum ^{n}_{j=1}y_{j}\dfrac{f\left( y_{j},x_{i},z_{k}\right) }{f\left( x_{i},z_{k}\right) }$
substituting the summation for the inner expectation:
$E\left( E\left( Y| Z,X\right) | X\right) = E\left( \sum ^{n}_{j=1}y_{j}\dfrac{f\left( y_{j},x_{i},z_{k}\right) }{f\left( x_{i},z_{k}\right) }| X\right)$
I'm am not sure how to proceed. Am I even starting with the right idea?
If I'm understanding correctly for
$E\left( Y| X\right) =E\left( E\left( Y| Z,X\right) | X\right)$
to be true I have to sum up the distribution of Y over Z conditional on X?