Assume that random variables $X$ and $Y$ are identically distributed and absolutely continuous. Suppose that $E[X^aY^b]=E[X^a]E[Y^b]$ for all $a,b$ natural numbers. Is it true that random variables $X$ and $Y$ are independent?
2026-03-27 13:17:27.1774617447
Independence from moments
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Yes. The hypothesis implies $Ef(X)g(Y) =Ef(X)Eg(Y)$ for all polynomials $f$ and $g$, hence for all bounded continuous functions $f$ and $g$ and this implies independence.