We start off in at the origin in a Cartesian coordinate system. We take $n$ steps (of unit length). Each step can be in any direction (any angle $0\leq\theta<2\pi$ uniformly).
Say, a random variable $C_n$ is the final coordinate after $n$ steps.
I would assume that $\mathbb{E}(C_n)=(0,0) : \forall n$. (And for $n=1$ this is easy to see.)
But I am wondering how one would calculate in general?
Say at time $t$ you move in direction $\theta_t$, then the coordinate at time $t+1$ is $C_{t+1}=C_{t}+(\cos\theta_t,\sin\theta_t)$, so it follows $C_{n}=(\sum_n^i\cos\theta_i, \sum_n^i\sin\theta_i)$
Given $\theta_i$ is taken independently from a uniform distribution, $E(C_n)=(nE(\cos\theta), nE(\sin\theta))=(0,0)$.