The Factorial Moments of the Generalized Bernoulli-Fibonacci Distribution
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The generalized Bernoulli-Fibonacci distribution describes the waiting time until k successive successes occur in a Bernoulli process. In the case of a symmetric Bernoulli process, when the counting starts at time 1, we prove that the factorial moments of this distribution are multiples of the terms of a subsequence of the generalized k-step Fibonacci numbers. As a result, when the counting starts at time k, the factorial moments of the Bernoulli-Fibonacci distribution are linear combi-nations of the aforementioned subsequence of the k-step Fibonacci numbers. Up to now, most authors used the second version of the Bernoulli-Fibonacci distribution, and thus have been unable to provide formulas for all the factorial moments of the distribution for a general k. To establish the main result in our paper, we first prove a number of identities involving the roots of unity of order k + 1 and the inverses of the roots of the characteristic polynomial of the k-step Fibonacci numbers.
Applied Mathematics | Mathematics
The Factorial Moments of the Generalized Bernoulli-Fibonacci Distribution.