On Some Schemes in Bernoulli Sequences of Random Variables

Bernoulli sequences of independent random variables X1, X2, … taking the value 1 with some probability 0 < p < 1 and the value 0 with the probability q = 1 – p are considered. The events {Xn = 1} and {Xn = 0} are treated, respectively, as “success” and “failure” in the nth Bernoulli trial. Works regularly appear that explore various situations related to the occurrence of such successes, failures, and series of them in Bernoulli sequences. In their earlier studies, the authors of this work obtained a number of new results for series of successes. Note that if the probabilities p and q are interchanged in the relations for the number of series of successes, similar results can be stated for series of failures. These earlier works studied properties of distributions of random variables such as the number of distinct success streaks among n variables, the number of distinct success streaks observed at the occurrence of the nth success, the number of success streaks at the occurrence of the mth failure, and the number of distinct success streaks observed at the occurrence of the first kth success streak. This work studies the properties of new random variables associated with the occurrence of various combinations of success and failure streaks in Bernoulli sequences.