### 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