### Expected Value

**Definition: **
The *expected value* of a continuous random variable
is denoted
and is defined by

(C.12) |

where denotes the

*probability density function*(PDF) for the random variable v.

**Example: **
Let the random variable
be uniformly distributed between
and
, *i.e.*,

(C.13) |

Then the expected value of is computed as

(C.14) |

Thus, the expected value of a random variable uniformly distributed between and is simply the average of and .

For a stochastic process, which is simply a sequence of random
variables,
means the expected value of
over
``all realizations'' of the random process
. This is also
called an *ensemble average*. In other words, for each ``roll of
the dice,'' we obtain an entire signal
, and to compute
, say, we average
together all of the values of
obtained for all ``dice rolls.''

For a stationary random process
, the random variables
which make it up
are identically distributed. As a result, we may normally compute
expected values by *averaging over time* within a *single
realization* of the random process, instead of having to average
``vertically'' at a single time instant over many realizations of the
random process.^{C.2} Denote time averaging by

(C.15) |

Then, for a stationary random processes, we have . That is, for

*stationary*random signals, ensemble averages equal time averages.

We are concerned only with stationary stochastic processes in this
book. While the statistics of noise-like signals must be allowed
to evolve over time in high quality spectral models, we may require
essentially time-invariant statistics within a single *frame* of
data in the time domain. In practice, we choose our spectrum analysis
window short enough to impose this. For audio work, 20 ms is a
typical choice for a frequency-independent frame length.^{C.3} In a multiresolution system, in which the frame length
can vary across frequency bands, several periods of the band
center-frequency is a reasonable choice. As discussed in
§5.5.2, the minimum number of periods required under
the window for resolution of spectral peaks depends on the window type
used.

**Next Section:**

Mean

**Previous Section:**

Stationary Stochastic Process