Continuous probability distributions

Suppose, now, 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,

$$P(u\in u:u+du) = P(u)\,du,$$ (23)

where $P(u)$ is known as the probability density. The earlier results (5), (12), and (19) generalize in a straight-forward manner to give

$$1 = \int_{-\infty}^\infty P(u)\,du,$$ (24)

$$\langle u\rangle = \int_{-\infty}^\infty P(u)\,u\,du,$$ (25)

$$\left\langle({\mit\Delta} u)^2\right\rangle = \int_{-\infty}^\infty P(u)\, (u-\langle u\rangle)^2\,du = \left\langle u^2\right\rangle-\langle u\rangle^2,$$ (26)

respectively.