With a PDF we can compute expectation values of selected quantities such as $$\begin{equation*} \langle x^k\rangle=\sum_{i=1}^{N}x_i^kp(x_i), \end{equation*}$$ if we have a discrete PDF or $$\begin{equation*} \langle x^k\rangle=\int_a^b x^kp(x)dx, \end{equation*}$$ in the case of a continuous PDF. We have already defined the mean value $\mu$ and the variance $\sigma^2$.