2.5:Continuous Probability Distributions
- Page ID
- 1188
Suppose, now, that the variable \(\begin{equation}u\end{equation}\) can take on a continuous range of possible values. In general, we expect the probability that \(\begin{equation}u\end{equation}\) takes on a value in the range \(\begin{equation}u \text { to } u+d u\end{equation}\) to be directly proportional to \(\begin{equation}d u\end{equation}\), in the limit that \(\begin{equation}d u \rightarrow 0\end{equation}\). In other words,
\begin{equation}P(u \in u: u+d u)=P(u) d u\end{equation}
where \(\begin{equation}P(u)\end{equation}\) is known as the probability density. The earlier results (5), (12), and (19) generalize in a straightforward manner to give
\begin{equation}\begin{aligned}
1 &=\int_{-\infty}^{\infty} P(u) d u \\
\langle u\rangle &=\int_{-\infty}^{\infty} P(u) u d u \\
\left\langle(\Delta u)^{2}\right\rangle &=\int_{-\infty}^{\infty} P(u)(u-\langle u\rangle)^{2} d u=\left\langle u^{2}\right\rangle-\langle u\rangle^{2}
\end{aligned}\end{equation}
respectively.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
\( \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}}\) \(\newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}}\) \(\newcommand {\btau}{\mbox{\boldmath$\tau$}}\) \(\newcommand {\bmu}{\mbox{\boldmath$\mu$}}\) \(\newcommand {\bsigma}{\mbox{\boldmath$\sigma$}}\) \(\newcommand {\bOmega}{\mbox{\boldmath$\Omega$}}\) \(\newcommand {\bomega}{\mbox{\boldmath$\omega$}}\) \(\newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}\)