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Last year, Nassim Taleb posted this on Twitter:

For a correlation number between X and Y, what is the "half point" correlation between 0 and 1, that is, the correlation for which you have half the uncertainty about X if you observed Y ( or vice-versa) compared to not having observed Y?

The answer is apparently 0.87 (specifically, $\sqrt{3}/2)$, and it apparently comes from Equation (13) here:

enter image description here

Can anyone explain what is going on?

How is $\sqrt{3}/2$ derived? And what does it actually mean, intuitively?

Thev
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  • There's a misprint in Equation 12, I think. It should say $X|Y ~ ...$ instead of $E(X|Y) ~ ...$ because $E(X|Y)$ was calculated in Equation 10. Anyway, the $\sqrt{3}/2$ is obtained by setting $\sqrt{(1-\rho^2)V(X)} = \sqrt{V(X)}/2$, in other words, it's the value of $\rho$ for which the standard deviation of $X|Y$ is half the standard deviation of $X$. – Flounderer Aug 21 '20 at 03:09
  • @Flounderer got it! My follow up question would be - what's the significance of this? – Thev Aug 21 '20 at 04:36
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    I'm not sure actually. Perhaps it is just saying that the result is counterintuitive, because a correlation of 0.87 is usually regarded as very high, so you might expect that knowing the value of Y, and knowing that Y is highly correlated with X, would make you a lot more certain about X. – Flounderer Aug 21 '20 at 09:10
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    For some intuition about the expression $\sqrt{1-\rho^2},$ see my post at https://stats.stackexchange.com/a/71303/919. – whuber Aug 21 '20 at 14:30

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