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Expectation Value

For one discrete variable,

\begin{displaymath}
\left\langle{f(x)}\right\rangle{} = \sum_x f(x)P(x).
\end{displaymath} (1)

For one continuous variable,
\begin{displaymath}
\left\langle{f(x)}\right\rangle{} = \int f(x)P(x)\,dx.
\end{displaymath} (2)

The expectation value satisfies
\begin{displaymath}
\left\langle{ax+by\rangle = a\langle x\rangle +b\langle y}\right\rangle{}
\end{displaymath} (3)


\begin{displaymath}
\left\langle{a}\right\rangle{} = a
\end{displaymath} (4)


\begin{displaymath}
\left\langle{\sum x}\right\rangle{} =\sum \left\langle{x}\right\rangle{}.
\end{displaymath} (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} (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} (7)

The (multiple) expectation value satisfies
$\displaystyle \left\langle{(x-\mu_x)(y-\mu_y)}\right\rangle{}$ $\textstyle =$ $\displaystyle \left\langle{xy-\mu_xy-\mu_yx+\mu_x\mu_y}\right\rangle{}$  
  $\textstyle =$ $\displaystyle \left\langle{xy}\right\rangle{}-\mu_x\mu_y-\mu_y\mu_x+\mu_x\mu_y$  
  $\textstyle =$ $\displaystyle \left\langle{xy}\right\rangle{}-\left\langle{x}\right\rangle{}\left\langle{y}\right\rangle{},$ (8)

where $\mu_i$ is the Mean for the variable $i$.

See also Mean




© 1996-9 Eric W. Weisstein
1999-05-25