Recurrence algorithms of waiting time for the success run of length $k$ in relation to generalized Fibonacci sequences

TL;DR

This paper develops recurrence algorithms for the distribution of waiting times for consecutive successes in Bernoulli trials, utilizing generalized Fibonacci sequences, and applies MLE for parameter estimation with simulations.

Contribution

It introduces recurrence algorithms based on generalized Fibonacci sequences for calculating distributions of waiting times in Bernoulli trials, including dependent cases.

Findings

Recurrence algorithms accurately compute waiting time distributions.

MLE methods effectively estimate parameters from simulated data.

Generalized Fibonacci sequences facilitate analysis of dependent Bernoulli trials.

Abstract

Let $V(k)$ denote the waiting time, the number of trials needed to get a consecutive $k$ ones. We propose recurrence algorithms for the probability distribution function (pdf) and the probability generating function (pgf) of $V(k)$ in sequences of independent and Markov dependent Bernoulli trials using generalized Fibonacci sequences of order $k$. Maximum likelihood estimation (MLE) methods for the probability distributions are presented in both cases with simulation examples.