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:  

$$m_x(t) \eq \E[x(t)]$$ (6)

The ``variance'' $\sigma^2$is defined to be the power after the mean is removed, i.e.,  

$$\sigma_x(t)^2 \eq \E\, [(x(t) - m_x(t))^2]$$ (7)

(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 x_t, 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,  

$$\E \quad \equiv \quad (\lim N \rightarrow \infty) \quad {1 \over N} \sum^N_1$$ (8)

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 (7):  

$$\sigma_x^2 \eq \E\, [(x_t - m_x)^2] \eq \E(x_t^2) \ - \ 2 m_x \E(x_t) + m_x^2 \eq \E(x_t^2) \ - \ m_x^2$$ (9)

Equation (9) says that the energy is the variance plus the squared mean.