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.