I was reading about generating random unit vectors for n-dimensional space.
I read one method in which the person suggested that we can take n-random samples from standard normal distribution and if we normalize it then we will get a random vector. The reason he gave was that Gaussian distribution is spherically symmetric.
Then I thought of another method , If we pick values from uniform distribution from $-\infty$ to $+\infty$ and then also after normalizing it, We will get a random vector. Since,probability for each point to be getting selected is same then probability for each random vector will be also same.
So ,I was wondering whether what I thought is correct or not? and also what are all distributions for which this will work and if normal distribution is the only distribution for which it will work then what is the reason behind it?