0

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?

Piotr100
  • 1

1 Answers1

1

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.

joriki
  • 238,052
  • Thank you very much. I hope that the following stronger property E[X^a Y^b]=E[X^a]E[X^b], which holds for all natural number a and b is still weaker than independence X of Y. Is it true? – Piotr100 Jul 04 '18 at 07:06
  • @Piotr100: I think you meant $E[X^aY^b]=E[X^a]E[Y^b]$? I suspect that that might actually be enough for independence. This is because you can use these powers for power series, and linearity of expectation then yields that the entire power series are uncorrelated, and you can approximate delta impulses with power series; and if any pair of delta impulses can be written as a product, then so can the entire distribution. But I'm not sure about this; there are others on the site who might be better at answering that question -- you could post it as a new question (possibly referencing this one). – joriki Jul 04 '18 at 07:49