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Are there any random variables so that E[XY] exists, but E[X] or E[Y] doesn't?

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What about a trivial case where $X=0$ and $Y$ has no expected value? –  Asaf Karagila Apr 9 '11 at 17:23
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Do you know someone called Tzwick who just asked "Are there any random variables so that E[X] and E[Y] exist but E[XY] doesn't?" –  Henry Apr 9 '11 at 17:42
    
@Henry: the provided e-mails are the same and both are unregistered. I've merged the accounts. @Tzwick: to stop from making more duplicate accounts, please register your account. –  Qiaochu Yuan Apr 9 '11 at 17:50
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2 Answers

Yes. For example take $C$ as a Cauchy random variable and independently $H$ as $0$ or $1$ with equal probability.

Let $X=CH$ and $Y=C(1-H)$.

Then the expectations of $X$ and $Y$ would be half the expectation of $C$, except that it does not exist, while $XY=0$ and so $E[XY]=0$.

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Sure, let Y be the value $2^n$ with probability $\frac{1}{2^n}$. Let $X=1/Y$. Then $XY=1$ and $E(X)=\frac{4}{3}$ exists, but $E(Y)$ does not exist.

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$Y$ definition is not complete. From your definition it is clear that $P(Y=2^n)=\frac{1}{2^n}$, but this leaves $1-2^{-n}$ of probability mass unassigned. –  mpiktas Apr 11 '11 at 12:35
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