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I have a very quick question regarding the expected value of two random variables $X,Y$ that are identically distributed and not necessarly independent.

Is this equation valid?

$E[XY]=E[X^2]$

If this is not true, why is this? and what relations can I get (regarding expected value, variance and covariance) when two random variables are identically distributed?

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  • $\begingroup$ Have you tried calculating $E[XY]$ and $E[X^2]$ for some simple examples? Try $X \sim N(0,1)$ and $Y = -X$. $\endgroup$ – JimmyK4542 Aug 29 '14 at 4:43
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No, this equation is not valid. To see this, consider a simple example in which random variable $X$ is either $0$ or $1$, each with probability 1/2. Random variable $Y$ has the same distribution, but is perfectly negatively correlated with $X$. Therefore, $X^2$ has the same distribution as $X$, and $E[X^2] =1/2$. However, $E[XY]=0$ since whenever $X=0$, $Y=1$ and whenever $X=1$, $Y=0$, so $XY=0$.

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  • $\begingroup$ Thanks for your answer, this is a great explanation :) $\endgroup$ – Pablo Estrada Aug 29 '14 at 4:47
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By definition:

$$cov(X, Y) = E(X-EX)(Y-EY) = EXY - (EX)(EY).$$ If $X, Y$ are independent, then $cov(X, Y) = 0,$ so that $EXY = (EX)(EY).$

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