How can I calculate a conditional expectation like below?
$E(X^3 - Y^3 | X-Y)$
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$$E(X^3 - Y^3 | X-Y)= E(X^3 | X - Y) - E(Y^3 | X - Y)$$
You never said if the variable was discrete or continuous. I'm going to do the discrete case below.
$$=\sum(X^3 \times \textrm{Pr}(X^3|X-Y)) - \sum(Y^3 \times \textrm{Pr}(Y^3|X-Y)) $$
For the continuous case, you integrate where there's a summation, and use the PDF for the conditional probability.
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1$\begingroup$ Thank you.I meant the continuous but it also will be useful for discrete mod. $\endgroup$ Commented Oct 25, 2019 at 13:54