1.4: Continuous Probability Distributions

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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)

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