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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? – cardinal Sep 18 '11 at 21:55
no, I am trying to think up examples where correlation implies independence – cbear Sep 18 '11 at 22:32
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 $Y$. – Luke Sep 18 '11 at 22:39
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. :) – cardinal Sep 18 '11 at 23:12

More generally, suppose $X$ and $Y$ are discrete random variables taking $m$ and $n$ possible values respectively. Then $X$ and $Y$ are independent iff $X^j$ and $Y^k$ are uncorrelated for $j=1 \ldots m-1$ and $k = 1 \ldots n-1$. The reason is that $\{X^j: j=0 \ldots m-1\}$ and $\{Y^k: k=0 \ldots n-1\}$ are bases of the vector spaces of functions of $X$ and $Y$ respectively.

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