Given two uniformly random directions on a hemisphere, n0 and n1, is the normalized sum of these vectors also a uniformly random direction on the same hemisphere?
Is average of two random directions also a random direction?
3 Answers
No - average directions away from the edge are more likely than average directions near the edge of the hemisphere
@Henry gave you a good answer, I just wanted to supplement it with some visuals. Using normalized standard Gaussian 3D vector to produce points, uniformly distributed on $S^2$ (see near the end of this article on MathWorld) in Mathematica:
The above visualizes distribution for the normalized sum of $n=1$, $n=2$, $n=4$ and $n=8$ vectors uniformly distributed on a hemisphere. Computation of the probability for the $z$-component of such a random point to be above $1/2$ then follows and shows greater concentration of points near then north pole as $n$ increases.
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0*Mag$n$ifique*. :-) – 2012-05-24
If you roll one die, the data is uniformly distributed. How about with 2 dice? It's not uniform anymore, because options closer to the centre of the data can be reached in more ways. Convolution is your friend.
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0I agree that the sum can be uniform in some cases, but I that's why I didn't draw the analogy with them. I was hoping that the bit about reaching some data in more ways would give the intuition as to the reason. – 2012-05-25