a******h
发帖数: 1183 | 1 刚刚胡思乱想想到的，不过还没想通。
两个随机变量，X,Y，假设他们有一样的distribution，他们的最大值的期望值 E[max(
X,Y)]是不是应该在X和Y独立的情况下最大呢？
我的想法：
当X,Y完全相关的时候，即X=Y，这时E[max(X,Y)]=E[X]，这样的话，当他们独立的时候
或者部分相关的时候E[max(X,Y)]一定是>=E[X]的。 | Q***5
发帖数: 994 | 2 X = 1 or -1 with prob 0.5 for each.
Then E(max(X,Y)) is maximized when Y = -X | a******h
发帖数: 1183 | 3 You're right.
I guess if X and Y are positive correlated, then the more they are related,
the less E[max(X,Y)] is, right?
【在 Q***5 的大作中提到】
: X = 1 or -1 with prob 0.5 for each.
: Then E(max(X,Y)) is maximized when Y = -X
| s******h
发帖数: 539 | 4 I would suppose we all assume that first moment of X exists, then
E(max(X,Y)) = EX + E|X - Y|/2 > = EX,
and how large is E(max(X,Y)) is equivalent to say 'How large is E|X - Y|'.
Clearly, E|X - Y| <= E|X| + E|Y| = 2 E|X|, that is, it reaches maximum value
if and only if |X| + |Y| = |X - Y| a.s., i.e. X + Y = 0. |
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