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I'm reading chapter 4 of DeGroot and either I missed it or he didn't explain it but I am not seeing how, given a joint distribution f(X, Y), how do we find E[X] and E[Y]. I have learned how to find E[r(X, Y)] where r(X, Y) is a real function involving X and Y. But I don't know how to find E[X] and E[Y] give f(X, Y). Do we just treat one or the other as a constant and just $\int x f(X, Y) dx$?

Thanks.

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Your suggested approach would find the conditional mean of, in this case, $X$ given $Y$, typically written $\mathbb{E}[X|Y]$. In order to find the marginal mean, we need to integrate $Y$ out of the expression $\mathbb{E}[X|Y]$ (this is known as the Law of Iterated Expectation, see for example https://brilliant.org/wiki/law-of-iterated-expectation/.)

This integration can be written:

$$\begin{split} \mathbb{E}[X] = \mathbb{E}_Y[\mathbb{E}[X|Y]] &=& \int_Y\mathbb{E}[X|Y=y]f(y)dy \\ &=& \int_Y\left[\int_Xxf(x|y)dx \right] f(y)dy\\ &=&\int_Y\int_Xxf(x,y)dxdy \end{split}$$

where the last step follows from $f(x,y) = f(x|y)f(y)$.

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