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The wikipedia definition of correlation is $$ \rho_{X,Y} = \frac{\mathrm{Cov(X,Y)}}{\sigma_X\sigma_Y} $$ where $X,Y$ are random variables with standard deviations $\sigma_X, \sigma_Y$ respectively. Of course this is nonsense if $X=0$ almost surely. Just to make sure, we define $\rho_{X,Y} =0$ in this case, correct? Or is there a broader definition of $\rho_{X,Y}$ which does not involve division by $\sigma_X$?

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I have only encountered the correlation between non-degenerate random variables. The definition i'm aware of:

Let $X$ and $Y$ be real non-degenerate random variables from $\mathcal{L}^2(P)$. Then the correlation between $X$ and $Y$ are defined to be $$ \rho(X,Y)=\frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}. $$

Note that $X$ is degenerate if and only if $\mathrm{Var}(X)=0$. I suppose the important property of the correlation is the following:

Let $X$ and $Y$ be non-degenerate random variables from $\mathcal{L}^2(P)$. Then the following are equivalent:

  1. $\rho(X,Y)\in\{-1,1\}$,

  2. There exists $\alpha,\beta\in\mathbb R$ such that $Y=\alpha+\beta X$ a.s.

By allowing degenerate random variables into the definition with $\rho=0$, the above result is no longer true. If $X=c_1$ and $Y=c_2$ a.s. then $Y=\alpha+\beta X$ a.s. with $\alpha=0$ and $\beta=\frac{c_2}{c_1}$. I would probably just stick with the definition for non-degenerate variables.

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