Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

6.11 RANDOM SEQUENCES

Now that we have described some basic properties of random processes, we consider important cases with specific distributions and properties. Discrete-time random sequences are covered in this section, and continuous-time random processes are described in Section 6.12.

6.11.1 Bernoulli Sequence

We begin with a formal definition of the Bernoulli random sequence that was discussed in Example 6.6 as a model for repeated coin tosses.

Definition: Bernoulli Sequence Bernoulli sequence X[k] is a set of iid random variables with outcomes {0, 1}, where P(X[k] = 1) = p and .

This sequence is strictly stationary. Each random variable can be viewed as a trial of which there are two outcomes. Outcome 1 is referred to as a success while 0 is called a failure. The expectation of a Bernoulli sequence is , the second moment is , and the variance is var[X[k]] = p−p2 = pq. These, of course, are the same results for the Bernoulli random variable because the pdf does not change with time in the definition above.

We can also examine related events that are modeled by random variables ...