On waiting time distribution of runs in a Fibonacci and Lucas sequences

Abstract

There is an increasingly needed to development a new mathematical apparatus concerned with the description, prediction, and understanding of natural phenomena in a precise manner. The purpose of this study is to extend the mathematical framework of Fibonacci and Lucas sequences for underpinned and establishing a modern mathematical formulas. For this purpose, we derive analytical formulas of order k-Fibonacci and order k-Lucas sequences simultaneously based on Bernoulli sequence. Furthermore, exploiting a relationship with the k th order Fibonacci and Lucas sequence, we study the probability distribution function (pdf) of the waiting time (W (k)).

Keywords

• Bernoulli trials
• Order k - Fibonacci sequence
• Order k -Lucas sequence

Bir Fibonacci ve Lucas dizisinde tekrarların bekleme süresi üzerine

Öz

Doğal fenomenlerin kesin bir manada tanımlanması, öngörülmesi ve anlaşılmasıyla ilgili yeni matematiksel aygıtlar geliştirme ihtiyacı giderek artmaktadır. Bu çalışmanın amacı, Fibonacci ve Lucas dizilerinin matematiksel çerçevesini modern matematiksel formüller ile desteklemek amacıyla genişletmektir. Bu amaçla, order-k Fibonacci ve order-k Lucas dizileri ile eş zamanlı olarak Bernoulli dizisine dayanarak analitik formüllerin türetilmesi amaçlanmaktadır. Ayrıca, k-yıncı mertebeden Fibonacci ve Lucas dizileri arasındaki bir ilişkiden yararlanarak (W (k)) bekleme süresinin olasılık dağılım fonksiyonu (pdf) çalışılacaktır.

Anahtar Kelimeler

• Bernoulli denemeleri
• k-mertebeli Fibonacci dizisi
• k-mertebeli Lucas dizisi

Kaynakça

1. Koutras, M. V. (1996). On a waiting time distribution in a sequence of Bernoulli trials. Annals of the Institute of Statistical Mathematics, 48(4), 789-806.
2. Chaves, L. M., & de SOUZA, D. J. (2007). Waiting time for a run of N successes in Bernoulli sequences. Rev. Bras. Biom, 25(4), 101-113.
3. Aki, S., & Hirano, K. (2007). On the waiting time for the first success run. Annals of the Institute of Statistical Mathematics, 59(3), 597-602.
4. Kim, S., Park, C., & Oh, J. (2013). On waiting time distribution of runs of ones or zeros in a Bernoulli sequence. Statistics & Probability Letters, 83(1), 339-344.
5. Öcal, A. A., Tuglu, N., & Altinişik, E. (2005). On the representation of k-generalized Fibonacci and Lucas numbers. Applied mathematics and computation, 170(1), 584-596.
6. Bekker, B. M., Ivanov, O. A., & Ivanova, V. V. (2016). Application of Generating Functions to the Theory of Success Runs. Applied Mathematical Sciences, 10(50), 2491-2495.
7. Singh, B., Sisodiya, K., & Ahmad, F. (2014). On the Products of-Fibonacci Numbers and-Lucas Numbers. International Journal of Mathematics and Mathematical Sciences, 2014.
8. Greenberg, I. (1970). The first occurrence of n successes in N trials. Technometrics, 12(3), 627-634.

9. Klots, J. H., & Park, C. J. (1972). Inverse Bernoulli Trials with Dependence (No. UWIS-DS-72-311). WISCONSIN UNIV MADISON DEPT OF STATISTICS.
10. Saperstein, B. (1973). On the occurrence of n successes within N Bernoulli trials. Technometrics, 15(4), 809-818.
11. Koutras, M. V. (1996). On a waiting time distribution in a sequence of Bernoulli trials. Annals of the Institute of Statistical Mathematics, 48(4), 789-806.
12. Koutras, M. V. (1997). Waiting times and number of appearances of events in a sequence of discrete random variables. In Advances in combinatorial methods and applications to probability and statistics (pp. 363-384). Birkhäuser Boston.