Can two uncorrelated variables have some relationship? If so, could you give me some examples?
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Perhaps: $P[X=0]=P[X=1]=P[X=-1]=1/3$; $Y=\cases{0,& if $X\ne0$\cr 1, &if $X=0$}$. – David Mitra Jan 23 '14 at 15:34
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Hint: Let $X$ take on values $-1$ and $1$, each with probability $\frac{1}{2}$, and let $Y=X^2$.
André Nicolas
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I am thinking about the formula: Cov (X,Y) = E[XY] - E[X]E[Y]...I know E[X] = 1/2(-1) * 1/2*(1) = 0...I forgot How to find E[XY]. Could you help this? – afsdf dfsaf Jan 23 '14 at 16:17
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Let $S$ and $X$ be independent random variables with $P\left[S=1\right]=P\left[S=-1\right]=\dfrac{1}{2}$, $E\left[X^{2}\right]<\infty$ and (for convenience) $E\left[X\right]=0$. Now define $Y:=XS$. Then $Y$ and $X$ are not independent. However their covariance is $0$ so they are uncorrelated. This because $E\left[XY\right]=E\left[X^{2}S\right]=E\left[X^{2}\right]E\left[S\right]=E\left[X^{2}\right]\times0=0$
drhab
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