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Gaussian Joint Variable Theorem

Also called the Multivariate Theorem. Given an Even number of variates from a Normal Distribution with Means all 0,

\begin{displaymath}
\left\langle{x_1x_2}\right\rangle{} = \left\langle{x_1}\right\rangle{}\left\langle{x_2}\right\rangle{},
\end{displaymath} (1)


\begin{displaymath}
\left\langle{x_1x_2x_3x_4}\right\rangle{} = \left\langle{x_1...
...gle{x_1x_4}\right\rangle{}\left\langle{x_2x_3}\right\rangle{},
\end{displaymath} (2)

etc. Given an Odd number of variates,
\begin{displaymath}
\left\langle{x_1}\right\rangle{} = 0,
\end{displaymath} (3)


\begin{displaymath}
\left\langle{x_1x_2x_3}\right\rangle{} = 0,
\end{displaymath} (4)

etc.




© 1996-9 Eric W. Weisstein
1999-05-25