Expected Value of Identically distributed random variables

Question

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?

Answer 1

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 $XY=0$.

Answer 2

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 $EXY = (EX)(EY).$