# 4 Random Processes

## 4.1 DEFINITION OF A RANDOM PROCESS

From Chapters 2 and 3 we learned that an experiment is specified by the three tuple (S, BF, P(·)), where S is a countable or noncountable set representing the outcomes of the experiment, BF is a Borel field specifying the set of events for which consistent probabilities exist, and P(·) is a probability measure that allows the calculation of the probability of all the events in the Borel field. A real random variable X(ξ) was defined as a real-valued function of S subject to (a) the event specified by {ξ : X(ξ) ≤ x} is a member of the BF for all x, thus guaranteeing the existence of the cumulative distribution, and (b) P{ξ : X(ξ) = ∞} = 0 or P{ξ : x(ξ) = −∞} = 0, or both.

If, instead of assigning a real value to each ξ, a time function X(t, ξ) is defined for each ξ, we say a random process is specified. Roughly speaking, a random process is a family of time functions together with a probability measure. For a finite sample space S we can visualize the random process as in Figure 4.1, that of a mapping from the sample space to a space of time waveforms.

A random process can also be viewed as shown in Figure 4.2. For a particular outcome ξi, with probability Pi, the time waveform shown X(t, ξi) occurs. The n time signals represent an ensemble of time waveforms.

If we evaluate these time waveforms at t0, the values on the second column from the right are obtained. Coming down that column of values, we have a mapping from the outcomes ...

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