Covariance of Order Statistics from N(0,1) Distribution


Actually, based on the demonstration graph of the integral region of phi shown upstairs, it should be from pi/4 to pi, because the shared point of cot(phi) and sqrt(2)*sin(theta) is pi/4. However, if we base on this integral bounds ( pi/4 < phi < pi), we cannot get the correct result.

The used integral bound of phi upstairs can work. however, it is short of a rigorous mathematical derivation. This is my pending question.

Hello, BGM,

The attached is my another approach to derive the integral bounds of phi in this topic. Although the integral result is correct using this derived bounds, I am not quite sure whether the derivation is correct or not.

You are expert on statistical mathematics, your help is greatly appreciated.

Thanks a lot.
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Hello Dragan and BGM

Eventually, I proved the integral bounds of phi in spherical coordinates by using a simple trangle and the graph of cot(x).Please see attached for details.

Initially, I was trying to use the comppound inequality to derive the said integral bounds, but it cannot work.

Many thanks for the help from Dragan and BGM in this topic and my previous ones.

Best Regards

DHB10 from China, PRC
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