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A conceptual average over the ensemble,
or **expectation**,
is denoted by the symbol .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 .The mean can vary with time:
| |
(12) |

The ``**variance**'' is defined to be the power after the mean is removed, i.e.,
| |
(13) |

(Conventionally, is referred to as the variance,
and is called the
``**standard deviation**.'')
For notational convenience, it is customary to write
*m*(*t*), , and *x*(*t*) simply as
*m*, , and *x*_{t},
using the verbal context to specify whether
*m* and 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 behaves like a summation sign, namely,

| |
(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 , we
can reduce equation (13):
| |
(15) |

Equation (15)
says that the energy is the variance plus the squared mean.

** Next:** Probability and independence
** Up:** TIME-STATISTICAL RESOLUTION
** Previous:** Ensemble
Stanford Exploration Project

10/21/1998