Intuition question. You draw two vectors from an n-dimensional isotropic Gaussian where n>>1. What is the typical angle between them?
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If the Gaussian is centered at zero, won’t both angles be uniformly distributed on the circle, and thus have undefined expected value? And doesn’t this imply the expectation of the difference is undefined?
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(For the case of n = 2.)
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yes. but n>>1 ;)
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NO!! Pick the first vector, call it its direction the z-axis. Pick a second vector. It has forms an angle theta with the z axis. You are being asked what is the mean value of theta. Use solid angles. The mean value is also the value with highest probability
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Yes - I confused (for n = 2) "angle between" (which has expectation pi/2) with "angle difference," for which the expectation does not exist.
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