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If we are given that two random variables $X$ and $Y$ are independent, I'm wondering if the rule: $E[XY] = E[X]E[Y]$ applies for any integer $k>0$, such that:

$E[X^kY^k] = E[X^k]E[Y^k]$.

Is this a straight forward result? or am I missing something fundamental?

Thanks for your comments / suggestions.

kentropy
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1 Answers1

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If $X$ and $Y$ are independent, so are $f(X)$ and $g(Y)$ where $f$ and $g$ are measurable.

Hence $X^k$ and $Y^k$ are independent. Hence, provided that they exist,

we have $$E[X^kY^k]=E[X^k]E[Y^k]$$

Siong Thye Goh
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  • Great. Thanks for your quick reply. I didn't consider the independence of the distributions. Great! Thanks – kentropy Apr 23 '18 at 06:14
  • How about the converse? Under what conditions can we infer that $X$ and $Y$ are independent if we know $E[X^k Y^\ell] = E[X^k] E[Y^\ell]$ for all $k, \ell$? (It's probably not wise to require $k = \ell$ here. – John C. Baez Dec 19 '19 at 20:47
  • That's an interesting question, unfortunately, I don't know the result yet. – Siong Thye Goh Dec 20 '19 at 03:28