consider a $x$ step random walk starting from origin in $n$-dimensional space where each step is taken into a random direction and has a distance of 1, i.e., each step is a vector on the $n$-dimensional unit sphere.
What is the mean and the variance of the squared euclidean distance from the origin after $x$ steps?
Empirically and with some stochastic juggling, I was able to deduce the following properties:
$\mu = x$
$\sigma^2 = 2x^2/n$
However, I need these properties for a paper and thus have to prove them. My "juggeling" is not a good proof and it is too lengthy to be included in the paper. So is there a short proof for these properties or can you give me a reference where this is proven (then I would just cite it)?