Suppose $X,Y$ are uncorrelated random variables, $\mathbb{E}(XY)=\mathbb{E}(X)\mathbb{E}(Y)$, taking on two values $m,n\in\mathbb{R}$, that is, $P(X\in \{m,n\})=P(Y\in \{m,n\})=1$. How should I go about showing that $X$ and $Y$ are indeed independent?
I think I can transform $X, Y$ into Bernoulli and show those two newly defined random variables are independent. Does this work?
I am thinking of setting $\zeta=\frac{X-m}{n-m}$, $\eta =\frac{Y-m}{n-m}$ and show these are independent