RANDOM SEQUENCES ( probability and statistic)

 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.

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.