We have four random variables say W,X,Y,Z where W and X has the same distribution and Y, Z also has the same distribution. Bad news is EX and EY may not exist but E(W+Z) is zero. Could we conclude that E(X+Y) is zero? ( I know if EX and EY where defined we used linearity and it is obvious, also we know nothing more about X, Y)



Expectation where linearity does not hold
Consider the following example. $W, X, Y, Z$ all have standard Cauchy distribution, but $X$ and $Y$ are independent while $Z = -W$. Then $X+Y$ also has a Cauchy distribution so its expected value does not exist, while $W+Z=0$ identically.


EDIT: With the additional assumption that $E[X+Y]$ exists, I believe you'll find that
$$ E[X+Y] = \int\_{-\infty}^\infty (F\_{-X}(t) - F\_{Y}(t))\ dt $$
(where $F\_{-X}$ and $F\_Y$ are the cumulative distribution functions of $-X$ and $Y$),
so that this is the same as $E[W+Z]$.


