1.4: Continuous Probability Distributions

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.

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

  $\newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}$ $\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}$ $\newcommand {\btau}{\mbox{\boldmath$\tau$}}$ $\newcommand {\bmu}{\mbox{\boldmath$\mu$}}$ $\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}$ $\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}$ $\newcommand {\bomega}{\mbox{\boldmath$\omega$}}$ $\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}$