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hi,

so I know E\[E\[X|Y\]\]=E\[X\]

how can I do:

E\[E\[XY|X\]\] ?

I did

E\[XE\[Y|X\]\]= XE\[E\[Y|X\]\] = XE\[X\]  
is that right?

​

Also in the same vien, how can I handle  
E\[X+X\^2\] using the law of total expectation?
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You have the "law" right (in scare quotes because this is just part of the definition of conditional expectation in measure-theoretic probability).  But you are messing up the rest.  You can pull X out of E(X Y | X) but you cannot pull X out of the *unconditional expectation.  Similarly, the "law" is not going to help you with E(X + X^2).  If you cannot do E(X Y) and E(X + X^2) without any reference to conditional expectation, then you cannot do it with.

It is true though that if you know E(Y | X) that E(X Y) = E{ X E(Y | X) }
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I haven't seen `E[E[X|Y]]=E[X]` and it took me a second to realize the inner expression means `E[X|Y]` integrated over all Y. It might help if you broke down the other expressions like that to make them easier to understand
ago

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