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

Question

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

Answer 1

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 $E[XY]=0$.

Answer 2

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.