4.2: Ensemble Averages

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Page ID: 47240

Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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The other set of statistics we can compute are across the ensemble, but at a particular time. Set \(y_i = x_i(t_o)\) where \(t_o\) is a specific time. Then, considering again the six harmonics from above, we have

\begin{align} E(y) = \sum p_i y_i = \sum_{i=1}^6 \, \dfrac{1}{6} a \cos (i \omega_o t_o) \\[4pt] E(y^2) = \sum p_i y_i ^2 = \sum_{i=1}^6 \, \dfrac{1}{6} a^2 \cos ^2 (i \omega_o t_o). \end{align}

We can see from this simple example that in general, time averages (which are independent of time, but dependent on event) and ensemble statistics (which are independent of event, but dependent on time) are not the same. Certainly one could compute ensemble statistics on time averages, and vice versa, but we will not consider such a procedure specifically here.

The ensemble autocorrelation function is now a function of the time and of the delay:

\begin{align} R(t, \, \tau) \, &= \, E(x(t) x(t + \tau)) \quad \textrm{or} \\[4pt] R(t, \, \tau) \, &= \, E[ \{ x(t) - E(x(t)) \} \{ x(t + \tau) - E(x(t + \tau)) \} ]. \end{align}

The second form here explicitly takes the mean values into account, and can be used when the process has nonzero mean. The two versions are not necessarily equal as written.