Mean Squared Displacement of Biased Random Walk

Question

If $x_t=x_{t-1}+\mathcal{N}(\mu,\sigma)$ and $x_0=0$ what's the value of $\langle x_t^2\rangle$?

Answer 1

The means add regardless of any other assumptions. Assuming (as usual in random walks) that the increments are independent, the variances also add. Hence the variance at time $t$ is $\sigma^2t$, and the mean is $\mu t$. Now

$$\text{Var}(x_t)=E[x_t^2]-(E[x_t])^2=E[x_t^2]-\mu^2t^2=\sigma^2t$$

hence

$$E[x_t^2]=\sigma^2t + \mu^2t^2.$$