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In other words, if X = Y in distribution, is it true that EX = EY?

I think this must be true, but I've tried to prove it a few times and I always get stuck.

Thanks in advance for any hints or reference.

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up vote 4 down vote accepted

Yes, this is true.

The expectation value depends only on the measure $P^X$ on $\mathbb R$ induced by the random variable $X$, and $X$ and $Y$ are, by definition, equal in distribution if and only if $P^X=P^Y$.

You can find this here:

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Thank you very much! – Not Durrett Nov 26 '10 at 17:55

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