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This is problem 6.9 in Ross.

I have the joint pdf $$f(x,y) = 6/7(x^2+xy/2), 0 marginal pdf $f(x) = (6/7)x(2x+1)$ conditional pdf $$f(y|x) = f(x,y)/f(x) = (x+y/2)/(2x+1)$$ conditional expectation $$E(y|x) = (3x+16)/(6(2x+1))$$

Then my professor has $$E(xy) = E(xE(y|x)) = \int_0^1 E(y|x) f(x) dx$$

This is the part I don't understand. I thought $E(xE(y|x))$ would be $$\int_0^1 xE(y|x)dx$$ but instead of the $x$, she has $f(x)$.

However, in another example I have $$ E(Y|X) = \frac 12 x. $$

Then my professor has $$ E(XE(Y|X)) = E(\frac 12 x^2) $$ I don't understand why in one case you multiply by $x$ and in the other by $f(x)$

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    thanks for editing. i wasn't sure how to use mathematical notation2012-12-19
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    For some basic information about writing math at this site see e.g. [here](http://meta.math.stackexchange.com/questions/5020/), [here](http://meta.stackexchange.com/a/70559/155238), [here](http://meta.math.stackexchange.com/questions/1773/) and [here](http://math.stackexchange.com/editing-help#latex).2012-12-19

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