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Average length of a line in a box

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I'm trying to come up with a statistical test for judging the quality of a prng. It involves randomly plotting 2 n-dimensional points in a n-dimensional box with sides of length 1, and finding the average distance between two points. I know the average squared distance should be dimensions/6, but I'm looking to find the unsquared distance. For 1 dimension I've calculated the average distance to be 1/3, but for dimensions 2 and higher I haven't been able to integrate the equations. For 2 dimensions the data looks like the average distance should be pi/6. Does anyone know how I could calculate these values?

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The 8 they have should be good enough for me, since this prng test wouldn't be testing a very high number of dimensions anyway. Thanks for the links!

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