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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

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    Is this homework?2011-09-18
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    no, I am trying to think up examples where correlation implies independence2011-09-18
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    I don't think you need to transform into Bernoulli. Calculate the probability that $X=x$ given that $Y=y$; it has to be the same as without conditioning on $Y$.2011-09-18
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    I think that the question you pose in the comment is a more interesting one and would attract more interest from the math.SE community. That said, very similar questions have already been asked, so you might want to browse the associated answers. :)2011-09-18

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