With the probability distribution function $w(x,t)dx$ we can evaluate expectation values such as the mean distance $$\langle x(t)\rangle = \int_{-\infty}^{\infty}xw(x,t)dx,$$ or $$\langle x^2(t)\rangle = \int_{-\infty}^{\infty}x^2w(x,t)dx,$$ which allows for the computation of the variance $\sigma^2=\langle x^2(t)\rangle-\langle x(t)\rangle^2$. Note well that these expectation values are time-dependent.