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I'm trying to write a monte carlo simulation for an optical analysis.

Half the area of a sphere is within 60 degrees of the poles. Hence, I'm assuming half of randomly directed radiation should fall within 60 degrees of the poles, when radiation is generated at the center of the sphere.

I have tried 3 algorithms so far for generating a random angle, and I test each one by averaging all of the angles. It seems to me that the average angle of a truly random distribution should be 60 degrees because half the angles will fall below 60 degrees and half above 60 degrees in a uniform distribution. So far all 3 algorithms give me the same value ~57.3 when averaged over hundreds of millions of runs.

So I'm thinking, either the pseudo-random number generator on python is making the same mistake in each algorithm, or I'm not correct that the average will be 60 degrees. Could someone please confirm that the average of a hemispherical distribution of angles would be 60? THANKS!

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# What is the average of a random hemispherical distribution

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