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Basic Concepts of Random Processes

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Definitions  ::  Ergodicity  ::  Autocorrelations  ::  Sinusoidal  ::  Poisson  ::  Gaussian  ::  Properties  ::  Spectra  ::  Cases  ::  Random  ::  Independent

Uncorrelated and independent


\begin{definition}
Two random processes are {\bf uncorrelated} if $C_{XY}(t,s) = 0$ for all
$s,t \in T$.
\end{definition}

\begin{definition}
The random processes $X_t$ and $Y_t$ are {\bf independent}...
...playmath} and all $t_1,
\ldots, t_n, s_1, \ldots, s_m \in T$.
\end{definition}
Note: Independence implies uncorrelated. The converse is not true.
\begin{definition}
The random processes $X_t$ and $Y_t$ are {\bf jointly Gaus...
...n \Zbb^+$ and all $t_1, \ldots, t_n,
s_1, \ldots, s_m \in T$.
\end{definition}
For jointly Gaussian random processes, we can characterize by a mean vector and a covariance matrix. All f.d.d.s are determined by $\mu_x(t)$ , $\mu_y(t)$ , $R_X(t,s)$ , $R_Y(t,s)$ and $R_{XY}(t,s)$ .

For this case, it is true that uncorrelated implies independence.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Basic Concepts of Random Processes. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture6_11.htm. This work is licensed under a Creative Commons License Creative Commons License