We know that $E(X+Y) = E(X) + E(Y)$. But why is $E|X-Y|$ $\ne$ $E|X| - E|Y|?$
2026-03-30 13:52:53.1774878773
Distribution of Expectation function into a $|X-Y|$
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If $X \ge Y$, then
$$E|X-Y| = EX - EY$$
Otherwise
$$E|X-Y| = EY - EX$$
Neither of which are necessarily equal to $$E|X| - E|Y|$$