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Next: Exercises Up: Probability Theory Previous: Mean, Variance, and Standard

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,
\end{displaymath} (23)

where $P(u)$ is known as the probability density. The earlier results (5), (12), and (19) generalize in a straightforward manner to give
$\displaystyle 1$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty P(u) du,$ (24)
$\displaystyle \langle u\rangle$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty P(u) u du,$ (25)
$\displaystyle \left\langle({\mit\Delta} u)^2\right\rangle$ $\textstyle =$ $\displaystyle \int_{-\infty}^\infty P(u)  (u-\langle u\rangle)^2 du = \left\langle u^2\right\rangle-\langle u\rangle^2,$ (26)



Richard Fitzpatrick 2010-07-20