= Mean-square displacement of an isotropic planar random walk
{title2=$\mathbb E\|S_n\|^2=n$}
Independent planar unit steps with uniformly distributed direction have zero vector <expected value>. The <squared Euclidean norm> of their sum has expectation $n$, because each diagonal term is one and all cross terms vanish by <independence>. More generally, zero-mean independent steps with common second moment $s^2$ give mean-square displacement $ns^2$; nonzero drift adds a quadratic contribution.
Back to article page
