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gjm comments on The usefulness of correlations - Less Wrong
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Eh?
That's just not what correlation means. (If we have, say, X=0.6Y or X=100Y or X=0.0001Y, exactly in each case, then the correlation is 1. The correlation tells you nothing about the coefficient in the relationship.)
Presumably X and Y have been converted to canonical form with mean 0, sd 1.
Yes, that was an error. I was thinking of the case where X and Y are both normalised to have s.d. 1, in which case the regression line is indeed Y = cX, but that isn't the case here. In general, the line is Y = bcX/a where the standard deviations of X and Y are a and b.