Expected distance from the origin for 2-d real numbers random walk

Question

We start at the point (0, 0) on the real 2-d plane. In every step $i$ we randomly and independently generate the angle $α_i \in [0, 2π]$, and then move from our current position by the unit vector determined by the angle $α_i$ (with the axis OX). What is the expected square of the distance from the point (0, 0) after n steps?

Answer 1

More generally suppose your displacements are $d_i\in\mathbb{R}^2$ and that $d_i$ and $d_j$ are independent for $i\neq j$... then after all displacements are applied you're at $\sum_i d_i$, whose expected squared length is $$ E\left[\sum_i d_i \cdot \sum_j d_j\right]=\sum_{i,j}E[d_i\cdot d_j]=\sum_i E\left[\|{d_i}\|^2\right], $$ since the $i\neq j$ terms are zero by the assumption of independence. In this specific case, $\|d_i\|^2=1$ with certainty, so the expected squared distance from the origin after $N$ steps is just $N$.