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Fundamental Multidimensional Variables

Abstract

The joint PDF of a Gaussian vector \( {\rm X} \in N_n \left( {\bar {\rm X},\sigma ^2 } \right) \) with covariance matrix
$$ {\rm M}_X = E\left\{ {\left( {{\rm X} - \overline {\rm X} } \right)\left( {{\rm X} - \overline {\rm X} } \right)^t } \right\} $$
(3.1)
is given as
$$ p_X \left( x \right) = \frac{1} {{\left( {2\pi } \right)^{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \left| {M_X } \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}\exp \left[ { - \frac{1} {2}\left( {x - \overline {\rm X} } \right)^t M_{_X }^{ - 1} \left( {x - \overline {\rm X} } \right)} \right] $$
(3.2)
where |MX| denotes the determinant of MX. Letting
$$ \omega = \left[ \begin{gathered} \omega _1 \\ \omega _1 \\ \cdot \\ \cdot \\ \omega _n \\ \end{gathered} \right] $$
(3.3)
then the joint CF is given by
$$ \Psi _X \left( \omega \right) = \exp \left[ {j\omega ^t \overline {\rm X} - \frac{1} {2}\omega ^t {\rm M}_X \omega } \right] $$
(3.4)
For the special case of n = 2, \( \bar {\rm X} = 0 \), and the covariance matrix of (1.10), the joint moments are given as
$$ E\left\{ {X_1^{k_1 } X_2^{k_2 } } \right\} = \left\{ \begin{gathered} 0,k_1 + k_2 odd \hfill \\ \sigma ^{k_1 + k_2 } \sum\limits_{i = 0}^{\left\lfloor {{{k_1 } \mathord{\left/ {\vphantom {{k_1 } 2}} \right. \kern-\nulldelimiterspace} 2}} \right\rfloor } {\left( \begin{gathered} k_1 \hfill \\ 2i \hfill \\ \end{gathered} \right)\left( {1 - \rho ^2 } \right)^i \rho ^{k_1 - 2i} \left( {k_1 + k_2 - 2i - 1} \right)} !!\left( {2i - 1} \right)!!, \hfill \\ k_1 + k_2 even \hfill \\ \end{gathered} \right. $$
(3.5)
.

Keywords

Mathematical Modeling Probability Distribution Covariance Matrix Communication Network Stochastic Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science + Business Media, LLC 2002

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