On binomial distributions of order k. (English) Zbl 0638.60024

Ling, K. D.

Stat. Probab. Lett. 6, No. 4, 247-250 (1988).

A. N. Philippou, C. Georghiou and the reviewer [ibid. 1, 171- 175 (1983; Zbl 0517.60010)] introduced the geometric and negative binomial distributions of order k. The binomial distribution of order k was introduced and studied independently by K. Hirano [Fibonacci numbers and their applications, Pap. 1st Int. Conf., Patras/Greece 1984, Math. Appl., D. Reidel Publ. Co. 28, 43-53 (1986; Zbl 0601.62023)] and by A. N. Philippou and F. S. Makri [Stat. Probab. Lett. 4, 211- 215 (1986; Zbl 0594.62013)].
In this paper a new binomial distribution of order k is derived and studied. It is defined as the distribution of the rv \[ M_ n^{(k)}=\sum^{n-k+1}_{i=1}\prod^{i+k-1}_{j=1}X_ j, \] where \(X_ 1,X_ 2,...,X_ n\) is a random sample of size n drawn on a Bernoulli rv X. The mean, the variance and the moment generating function of this distribution are obtained.

Reviewer: G.Philippou

MSC:

60E99

Distribution theory

Keywords:

geometric and negative binomial distributions of order k; Fibonacci numbers; moment generating function

Citations:

Zbl 0517.60010; Zbl 0601.62023; Zbl 0594.62013

References:

[1]	Aki, S, Discrete distributions of order k on a binary sequence, Ann. inst. statist. math., 37, 205-244, (1985), (Part A) · Zbl 0577.62013
[2]	Aki, S; Kuboki, H; Hirano, K, On discrete distributions of order k, Ann. inst. statist. math., 36, 431-440, (1984), (Part A) · Zbl 0572.62018
[3]	Hirano, K, Some properties of the distributions of order k, () · Zbl 0634.60014
[4]	Philippou, A.N; Georghiou, C; Philippou, G.N, A generalized geometric distribution and some of its properties, Statist. probab. letters, 1, 171-175, (1983) · Zbl 0517.60010
[5]	Philippou, A.N; Makri, F.S, Successes, runs and longest runs, Statist. probab. letters, 4, 211-215, (1986) · Zbl 0594.62013