<s>[INST] Correlation Coefficient - $\rho(X,Y)$. If I have two aleatory variables
$$X=\begin{pmatrix}2&3&4&5&6&7&8&9&10&11&12\\ \frac{1}{36}&\frac{2}{36}&\frac{3}{36}&\frac{4}{36}&\frac{5}{36}&\frac{6}{36}&\frac{5}{36}&\frac{4}{36}&\frac{3}{36}&\frac{2}{36}&\frac{1}{36}\end{pmatrix}$$ and $$Y=\begin{pmatrix}2&3&4&5&6&7&8&9&10&11&12\\ \frac{1}{36}&\frac{2}{36}&\frac{3}{36}&\frac{4}{36}&\frac{5}{36}&\frac{6}{36}&\frac{5}{36}&\frac{4}{36}&\frac{3}{36}&\frac{2}{36}&\frac{1}{36}\end{pmatrix}$$ how can I calculate $$\rho(X,Y) \mbox{?}$$
$\rho$ -correlation coefficeint. 
thanks:)
 [/INST] You cannot find the correlation without additional information.  $X$ and $Y$ could be independent, in which case the correlation is $0$, or (in this instance) $X$ might always be equal to $Y$ so the correlation would be $1$, or $X$ might (again, in this instance) be equal to $14-Y$ (as $X$ goes from $2$ to $12$, then $14-X$ goes from $12$ to $2$), in which case the correlation is $-1$, or they could be related in more complicated ways, in which case the correlation could have any of a large (but in this case finite) number of possible other values between $1$ and $-1$.
Since the number of possible values of the correlation is in this case finite, one could say that there's enough information given to deduce at least something about the correlation.
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