For one discrete variable,
![\begin{displaymath}
\left\langle{f(x)}\right\rangle{} = \sum_x f(x)P(x).
\end{displaymath}](e_2635.gif) |
(1) |
For one continuous variable,
![\begin{displaymath}
\left\langle{f(x)}\right\rangle{} = \int f(x)P(x)\,dx.
\end{displaymath}](e_2636.gif) |
(2) |
The expectation value satisfies
![\begin{displaymath}
\left\langle{ax+by\rangle = a\langle x\rangle +b\langle y}\right\rangle{}
\end{displaymath}](e_2637.gif) |
(3) |
![\begin{displaymath}
\left\langle{a}\right\rangle{} = a
\end{displaymath}](e_2638.gif) |
(4) |
![\begin{displaymath}
\left\langle{\sum x}\right\rangle{} =\sum \left\langle{x}\right\rangle{}.
\end{displaymath}](e_2639.gif) |
(5) |
For multiple discrete variables
![\begin{displaymath}
\left\langle{f(x_1,\ldots,x_n)}\right\rangle{} = \sum_{x_1, \ldots, x_n} f(x_1,\ldots ,x_n)P(x_1,\ldots ,x_n).
\end{displaymath}](e_2640.gif) |
(6) |
For multiple continuous variables
![\begin{displaymath}
\left\langle{f(x_1,\ldots,x_n)}\right\rangle{} = \int f(x_1, \ldots, x_n)P(x_1,\ldots ,x_n)\,dx_1\cdots\,dx_n.
\end{displaymath}](e_2641.gif) |
(7) |
The (multiple) expectation value satisfies
where
is the Mean for the variable
.
See also Mean
© 1996-9 Eric W. Weisstein
1999-05-25