Given two centered and scaled random variables $X$ and $Y$, can you relate the probability they have the same sign to their correlation? If the correlation is close to $1$, I am picturing the joint distribution in the plane to be concentrated mostly in the 1st quadrant and the 3rd quadrant, where $X$ and $Y$ have the same sign. Is that always the case? Is there a counter-example where the correlation is close to $1$ but the probability of having the same sign is low? Os is there a bound?
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1$\begingroup$ Yes, there are counterexamples. See stats.stackexchange.com/questions/562748. The intuition it imparts is that because the (Pearson) correlation depends on the value whereas the signs are insensitive to the values, you can create extreme differences between the two concepts of "correlation." $\endgroup$– whuber ♦Commented Jul 3 at 19:56
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2$\begingroup$ @whuber I think your link went to the home page of the site. $\endgroup$– Peter FlomCommented Jul 3 at 20:59
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$\begingroup$ Thanks @PeterFlom. I fixed the link. $\endgroup$– whuber ♦Commented Jul 4 at 12:20
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