Timeline for What is the probability of $P(X>Y>0)$ where $X$ and $Y$ standard normal distribution with correlation $\rho$?
Current License: CC BY-SA 4.0
15 events
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| Sep 14, 2023 at 0:17 | history | edited | Ben | CC BY-SA 4.0 |
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| Sep 13, 2023 at 22:44 | history | edited | Zhanxiong | CC BY-SA 4.0 |
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| Sep 13, 2023 at 22:35 | comment | added | Dilip Sarwate | @whuber Zhanxiong has already answered your question regarding symmetry. I apologize for the typo in my displayed equation; that $-$ should have been a $=$. Oh well, comments can’t be edited this late. | |
| Sep 13, 2023 at 20:53 | comment | added | Zhanxiong | @whuber I guess Dilip means $P(X > 0, Y > 0) = P(X > 0, Y > 0, X > Y) + P(X > 0, Y > 0, Y > X)$, and the probabilities on the right hand side are identical to $P(X > Y > 0)$. | |
| Sep 13, 2023 at 20:37 | comment | added | whuber♦ | @Dilip Which symmetry, specifically? | |
| Sep 13, 2023 at 20:22 | answer | added | Zhanxiong | timeline score: 2 | |
| Sep 13, 2023 at 18:24 | comment | added | Dilip Sarwate | According to Abramowitz and Stegun 26.3.19, p 937, $$P(X>0, Y>0) - \frac 14 + \frac{\arcsin \rho}{2\pi}$$ and so, because of the symmetry, $P(X>Y>0)$ should be half that number. | |
| Sep 13, 2023 at 17:53 | comment | added | whuber♦ | I hope you will find it's equivalent to one of my two geometric descriptions ;-). | |
| Sep 13, 2023 at 17:34 | comment | added | Zhihao Xu | Thank you! I will read it now. By the way, What do you think about what I have done using Cholesky? | |
| Sep 13, 2023 at 17:07 | comment | added | whuber♦ | I would like to suggest that simply looking at the plots in my post at stats.stackexchange.com/a/71303/919 will enable you to write down a correct simple formula. All you need to is reverse the transformations to find where the wedge $x\gt y\gt 0$ originated; the probability will equal the angle divided by $2\pi.$ | |
| Sep 13, 2023 at 16:49 | history | edited | Zhihao Xu | CC BY-SA 4.0 |
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| Sep 13, 2023 at 16:48 | history | edited | Zhihao Xu | CC BY-SA 4.0 |
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| Sep 13, 2023 at 16:38 | history | edited | Zhihao Xu | CC BY-SA 4.0 |
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| S Sep 13, 2023 at 16:36 | review | First questions | |||
| Sep 13, 2023 at 17:20 | |||||
| S Sep 13, 2023 at 16:36 | history | asked | Zhihao Xu | CC BY-SA 4.0 |