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As I keep reading probability books, there are always some issues that no one considers.

For example,

for $\omega \in \Omega$ and $X$, $Y$ independent random variable we define $Z(\omega )=X(\omega )\cdot Y(\omega)$, So if $E[X]$ , $E[Y]$ , $E[Z]$ defined, we know that $E[X]\cdot E[Y]=E[Z]$.

But, I really curious whether there's a situation when $E[X]$, $E[Y]$ defined, but $E[X\cdot Y]$ ($E[Z]$) is $\infty$ or even Diverging? I wasnt able to think of an answer.

(Is it ok to post more than one question in the same day?)

Thanks again.

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    That's not possible with $X$ and $Y$ independent. Do you still mean to include independence as an assumption?2011-04-06
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    yes, this is what I thought about. How come it's not possible? how does one prove that?2011-04-06
  • 0
    The formula $E[XY]=E[X]E[Y]$, when $X$ and $Y$ are independent integrable random variables, can be proved using the Monotone Convergence Theorem. But this is not elementary...2011-04-07

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