**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 *ξ*_{i}, with probability *P*_{i}, the time waveform shown *X*(*t*, *ξ*_{i}) occurs. The *n* time signals represent an ensemble of time waveforms.

If we evaluate these time waveforms at *t*_{0}, 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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