On A Generalization Of Ling’S Binomial Distribution

Abstract

: In a sequence of n binary trials, distribution of the random variable M_n,k, denoting the number of overlapping success runs of length exactly k, is called Ling’s binomial distribution or Type II binomial distribution of order k. In this paper, we generalize Ling’s binomial distribution to Ling’s q-binomial distribution using Bernoulli trials with a geometrically varying success probability. An expression for the probability mass function of this distribution is derived. For q = 1, this distribution reduces to Ling’s binomial distribution. 

Keywords

• Binomial distribution of order k,Ling’s binomial distribution,q-distributions,runs

References

1. Charalambides, C.A., 2010, The q-Bernstein Basis as a q-Binomial Distributions, Journal of Statistical Planning and Inference, 140, 2184-2190.
2. Godbole, A.P., 1992 The Exact and Asymptotic Distribution of Overlapping Success Runs, Communications in Statistics-Theory and Methods, 21, 953-967.
3. Hirano, K., 1986, Some Properties of the Distributions of Order k, Fibonacci Numbers and Their Applications (eds. G.E. Bergum, A.N. Philippou, and A.F. Horadam), 43-53.
4. Ling, K.D., 1988, On Binomial Distribution of Order k, Statistics & Probability Letters, 6, 247-250.
5. Makri, F. S., Philippo, A. N., and Psillakis, Z. M., 2007, Polya, Inverse Polya, and Circular Polya Distributions of Order k for l-Overlapping Success Runs, Communications in Statistics-Theory and Methods, 36, 657-668.
6. Philippou, A.N. and Makri, F.S., 1986, Success Runs and Longest Runs, Statistics & Probability Letters, 4, 211-215.
7. Yalcin, F. and Eryilmaz, S., 2014, q-Geometric and q-Binomial Distributions of Order k, Journal of Computational and Applied Mathematics, 271, 31-38.