Expected value of two different random variables

Question

I am a bit confused. We have two different random variables $X,Y$ and another two random variables $G,F$, such that $G$ has the same distribution as $X$ and $F$ has the same distribution as $Y$.
Does \begin{align*} E(XY)=E(GF) \end{align*} hold?
Or do we need some independence assumptions?
Sorry for this silly qustion:)

Answer 1

You need information regarding the joint distribution.

Counterexample:

Suppose all $4$ variables are $\pm 1$ with equal probability.

Scenario $1$: They are pairwise independent. Then both $XY$ and $GF$ are also $\pm 1$ with equal probability so both expectations are $0$ (in particular, they are equal).

Scenario $2$: $X,Y$ are pairwise independent but $G=F$. Then $E(XY)=0$ as before but $GF$ is the constant $1$, hence has expectation $1$.