Assume that random variables X and Y are identically distributed and absolutely continuous. Suppose that $E[XY]=E[X]E[Y].$ Is it true that Random variables X and Y are independent?
2026-03-26 04:50:00.1774500600
independence of two identically dostributed random variables
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It doesn't. For instance, they could be standard normal distributions whose signs are independent but whose magnitudes are identical.
$E[XY]=E[X]E[Y]$ means that they're uncorrelated. That's a far weaker property than independence. It removes a single degree of freedom, whereas independence qualitatively reduces the complexity of the distribution from one function of two variables to two functions of one variable.