3 Random Process

Def:
A discrete random process is called purely random.
-A discrete random is purely random if it consists of random variables \(\{Z_t\}\), which are mutually and independent.
-Mean is constant
-Variance is constant.

\begin {align*} \varphi (\tau )&= Cov(X_t,X_{t+\tau })\\ &= 0, k= \pm 1, \pm 2,.........\\ \end {align*}

-Since the mean and the variance do not depend on time, the process is second order.

The auto correlation function for a purely random process is
\[ \rho (k) = \begin {cases} 1, &k = 0\\ 0, &k = \pm 1, \pm 2, .....\\ \end {cases} \]

\(\{Z_t\}\)- White noise

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