$$\require{cancel}$$
Suppose that the variable $$u$$ can take on a continuous range of possible values. In general, we expect the probability that $$u$$ takes on a value in the range $$u$$ to $$u+du$$ to be directly proportional to $$du$$, in the limit that $$du\rightarrow 0$$. In other words, $P(u\in u:u+du) = P(u)\,du,$ where $$P(u)$$ is known as the probability density. The earlier results (1.2.4), (1.3.4), and (1.3.11) generalize in a straightforward manner to give: \begin{aligned} 1&= \int_{-\infty}^\infty P(u)\,du,\\[0.5ex] \langle u\rangle &= \int_{-\infty}^\infty P(u)\,u\,du,\\[0.5ex] \left\langle({\mit\Delta} u)^2\right\rangle &= \int_{-\infty}^\infty P(u)\, (u-\langle u\rangle)^2\,du = \left\langle u^{\,2}\right\rangle-\langle u\rangle^2,\end{aligned} respectively.