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Expectation and variance

A conceptual average over the ensemble, or expectation, is denoted by the symbol $\E$.The index for summation over the ensemble is never shown explicitly; every random variable is presumed to have one. Thus, the true mean at time t is defined as ${m_x}(t)=\E(x_t)$.The mean can vary with time:  
 \begin{displaymath}
m_x(t) \eq \E[x(t)]\end{displaymath} (12)
The ``variance'' $\sigma^2$is defined to be the power after the mean is removed, i.e.,  
 \begin{displaymath}
\sigma_x(t)^2 \eq \E\, [(x(t) - m_x(t))^2]\end{displaymath} (13)
(Conventionally, $\sigma^2$ is referred to as the variance, and $\sigma$ is called the ``standard deviation.'')

For notational convenience, it is customary to write m(t), $\sigma(t)$, and x(t) simply as m, $\sigma$, and xt, using the verbal context to specify whether m and $\sigma$are time-variable or constant. For example, the standard deviation of the seismic amplitudes on a seismic trace before correction of spherical divergence decreases with time, since these amplitudes are expected to be ``globally'' smaller as time goes on.

When manipulating algebraic expressions, remember that the symbol $\E$ behaves like a summation sign, namely,  
 \begin{displaymath}
\E \quad \equiv \quad
(\lim N \rightarrow \infty) \quad {1 \over N} \sum^N_1\end{displaymath} (14)
Note that the summation index is not given, since the sum is over the ensemble, not time. To get some practice with the expectation symbol $\E$, we can reduce equation (13):  
 \begin{displaymath}
\sigma_x^2
\eq \E\, [(x_t - m_x)^2]
\eq \E(x_t^2) \ - \ 2 m_x \E(x_t) + m_x^2
\eq \E(x_t^2) \ - \ m_x^2\end{displaymath} (15)
Equation (15) says that the energy is the variance plus the squared mean.


next up previous print clean
Next: Probability and independence Up: TIME-STATISTICAL RESOLUTION Previous: Ensemble
Stanford Exploration Project
10/21/1998