Is average of two random directions also a random direction?

Question

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?

Answer 1

No - average directions away from the edge are more likely than average directions near the edge of the hemisphere

Answer 2

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.

Answer 3

@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.