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

Björn Friedrich
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    What about a trivial case where $X=0$ and $Y$ has no expected value? – Asaf Karagila Apr 09 '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 09 '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 09 '11 at 17:50

2 Answers2

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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 $E[XY]=0$.

Henry
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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.

Thomas Andrews
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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