Andreas N. Philippou

Applications of Fibonacci Numbers

This volume contains a selection of papers presented at the Fifth International Conference on Fibonacci Numbers and Their Applications. The topics covered include number patterns, linear recurrences, and the application of the Fibonacci Numbers to probability, statistics, differential equations, cryptography, computer science, and elementary number theory. Many of the papers included contain suggestions for other avenues of research. For those interested in applications of number theory, statistics and probability, and numerical analysis in science and engineering: 1993, 625 pp. ISBN 0-7923-2491-9 Hardbound Dfl. 320.00 / £123.00 / US

A generalized geometric distribution and some of its properties

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Successes, runs and longest runs

A generalized geometric distribution is introduced and briefly studied. First it is noted that it is a proper probability distribution. Then its probability generating function, mean and variance are derived. The probability distribution of the sum Yr of r independent random variables, distributed as generalized geometric, is derived. Finally, sufficient conditions are presented under which we can derive the limiting distribution of Yr - kr as r --> [infinity].

Fibonacci numbers and their applications

The probability distribution of the numbeer of success runs of length k ( >/ 1) in n ( [greater-or-equal, slanted] 1) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let Sn and Ln, respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability P(Ln [less-than-or-equals, slant] k Sn = r) (0 [less-than-or-equals, slant] k [less-than-or-equals, slant] r [less-than-or-equals, slant] n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of Xn, Ln(k) is established, where Xn, Ln(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k.

Success run statistics defined on an urn model

Fibonaccene.- On a Class of Numbers Related to Both the Fibonacci and Pell Numbers.- A Property of Unit Digits of Fibonacci Numbers.- Some Properties of the Distributions of Order k.- Convolutions for Pell Polynomials.- Cyclotomy-Generated Polynomials of Fibonacci Type.- On Generalized Fibonacci Process.- Fibonacci Numbers of Graphs III: Planted Plane Trees.- A Distribution Property of Second-Order Linear Recurrences.- On Lucas Pseudoprimes which are Products of s Primes.- Fibonacci and Lucas Numbers and the Morgan-Voyce Polynomials in Ladder Networks and in Electric Line Theory.- Infinite Series Summation Involving Reciprocals of Pell Polynomials.- Fibonacci and Lucas Numbers and Aitken Acceleration.- On Sequences having Third-Order Recurrence Relations.- On the Solution of the Equation G n = P(x).- Distributions and Fibonacci Polynomials of Order k, Longest Runs, and Reliability of Consecutive-k-Out-Of-n : F Systems.- Fibonacci-Type Polynomials and Pascal Triangles of Order k.- A Note on Fibonacci and Related Numbers in the Theory of 2 x 2 Matrices.- Problems on Fibonacci Numbers and Their Generalizations.- Linear Recurrences having almost all Primes as Maximal Divisors.- On the Asymptotic Distribution of Linear Recurrence Sequences.- Golden Hops Around a Circle.

On multiparameter distributions of order k

Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

On binomial and circular binomial distributions of orderk forl-overlapping success runs of lengthk

A multiparameter negative binomial distribution of order k is obtained by compounding the extended (or multiparameter) Poisson distribution of order k by the gamma distribution. A multiparameter logarithmic series distribution of order k is derived next, as the zero truncated limit of the first distribution. Finally a few genesis schemes and interrelationships are established for these three multiparameter distributions of order k. The present work extends several properties of distributions of order k.

Polya, Inverse Polya, and Circular Polya Distributions of Order k for l-Overlapping Success Runs

The number ofl-overlapping success runs of lengthk inn trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulas are derived for its probability distribution function in terms of multinomial and binomial coefficients respectively. A recurrence relation concerning this distribution, as well as its mean, is also obtained. Furthermore, the number ofl-overlapping success runs of lengthk inn Bernoulli trials arranged on a circle is presently considered for the first time and its probability distribution function and mean are derived. Finally, the latter distribution is related to the first, two open problems regarding limiting distributions are stated, and numerical illustrations are given in two tables. All results are new and they unify and extend several results of various authors on binomial and circular binomial distributions of orderk.

Asymptotic normality of the maximum likelihood estimate in the independent not identically distributed case

The Polya-Eggenberger sampling scheme is considered, i.e., a ball is drawn at random from an urn containing w white (success) balls and b black (failure) balls, its color is observed, and then it is returned to the urn along with s additional balls of the same color of the ball drawn. This scheme is repeated n times, and N n, k, l, s denotes the number of l-overlapping success runs of length k. If the balls drawn are arranged on a circle, the number of l-overlapping success runs of length k is denoted by . Finally, W r, k, l, s denotes the number of drawings according to the Polya-Eggenberger sampling scheme until the rth occurrence of the l-overlapping success run of length k. Polya, inverse Polya, and circular Polya distributions of order k for l-overlapping success runs of length k are introduced as the distributions, respectively, of N n, k,l, s , W r, k, l, s , and . These distributions include as special cases known and new distributions of order k. Exact formulae are derived for their probability distribution functions and means, which generalize several results on well-known distributions of order k. Asymptotic results of the new distributions are also given, relating them, respectively, to the binomial, negative binomial, and circular binomial distributions of order k for l-overlapping success runs of length k.

Exact reliability formulas for linear and circular m-consecutive-k-out-of-n:F systems

In this paper, we assume the existence and consistency of the maximum likelihood estimate (MLE) in the independent not identically distributed (i.n.i.d.) case and we establish its asymptotic normality. The regularity conditions employed do not involve the third order derivatives of the underlying probability density functions (p.d.f.s).