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Can anyone give an example (joint probability table) that two Bernoulli variables are uncorrelated but not independent?

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  • $\begingroup$ Let $X\sim U(-1,1)$. Let $Y=X^2$. The variables are uncorrelated but dependent $\endgroup$ Commented May 19 at 13:37
  • $\begingroup$ I want Bernoulli. $\endgroup$
    – roz
    Commented May 19 at 20:19
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    $\begingroup$ Luckily I got a good answer before the stupid admin closes my question. $\endgroup$
    – roz
    Commented May 20 at 7:51

2 Answers 2

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If $X$ and $Y$ are Bernoulli and uncorrelated, they are independent. Indeed $E(X)=p,\ E(Y)=q, E(XY)=pq$ imply $$\Pr(X=Y=1)=pq,\ \Pr(X=1)=p,\ \Pr (X=1,Y=0)=\Pr(X=1)-\Pr(X=1,Y=1)=p(1-q).$$ Similarly $\Pr(X=0,Y=1)=q(1-p)$ and finally $\Pr(X=Y=0)=(1-p)(1-q).$ Hence $X$ and $Y$ are independent.

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  • $\begingroup$ Thanks for your answer. It is helpful and answers my question. $\endgroup$
    – roz
    Commented May 20 at 7:49
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Let $X\sim U(-1,1)$. Let $Y=X^2$.
The variables are uncorrelated but dependent

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  • $\begingroup$ Please don't answer low quality questions. $\endgroup$
    – amWhy
    Commented May 19 at 23:30
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    $\begingroup$ To expand on @amWhy's comment, we request that you try to avoid answering questions which don't meet the standards of the site. Answering such questions encourages people to ask more questions which don't meet site standards. Instead, please help users to ask better questions---use the comments below a question to suggest ways in which further context can be provided. For details, see math.meta.stackexchange.com/q/33508 . $\endgroup$
    – Xander Henderson
    Commented May 19 at 23:50

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