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  1. Let $X, Y$ be random variables. If $\rho(X,Y) = a$ (Correlation), where $a \in (0,1)$, what can be said about the relationship between $X$ and $Y$? Is it true that $Y = bX + c + Z$, where $Z$ is a random variable? If it is true then how is $|Z|$ related to the correlation a?
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With no independence assumption, it is difficult to say anything else than $Y=bX+Z$ with $b=a\sigma(Y)/\sigma(X)$ for a random variable $Z$ such that $\mathrm{Cov}(Z,X)=0$.

For example $Z$ could be $Z=XT$ for any centered random variable $T$ independent on $X$. If $T$ is $t$ times a centered Bernoulli random variable with $t\geqslant0$, $|Z|=t|X|$ is not determined by $a$ and may be as large as one wants.

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If $c$ is chosen to make $bX +c$ the linear least mean-square error estimator for $Y$ (you have given the appropriate value for $b$ already) , then doesn't $Z = Y - (bX +c)$ also have mean $0$ and variance $\sigma_Y^2(1 - a^2)$ in addition to being uncorrelated with $X$? This gives a weak relationship between the distribution of $Z$ or $\vert Z \vert$ and the value of $a$. – Dilip Sarwate Oct 13 '11 at 13:07

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