Sonali Bhattacharya*
Symbiosis Centre for Management and Human Resource Development, (Deemed University), India
Submission: October 23, 2017; Published: January 17, 2018
*Corresponding author: Sonali Bhattacharya, Associate Professor, Symbiosis International (Deemed University), India, Tel: 91-20-22934304;
Email:
0039
Introduction
Distribution of runs and successions in various situations
have been under considerable studies due to their applications
to reliability theory of consecutive systems Grifitth [2];
Papastravridis & Sfkianakis [3], Sfkianakis, Kounias & Hillaris [4],
Papastravridis & Koutras [5] & Cai [6], start-up demonstration
tests [7], molecular biology [1], theory of radar detection, time
sharing systems and quality control [8-12]. There are different
ways of computations and enumeration of number of runs:
1. Feller [13] defined ways of counting the runs of exactly
length k as counting the number from scratch everytime
a run occurs. For example the sequence | SSS SFSSS SSS F
contains 3 success runs of length 3
2. Goldstein [7] proposed the distribution of the number
of success runs of at least length k until the n-th trial. In
this way of counting the number of runs of length 3 (or
more), in the above example contains 2 success runs of
length 3 (or more)
3. Schwager [14] and Ling [15,16] studied the distributions
on the number of overlapping runs of length k . In
the enumeration scheme | SSS SFSSS SSS F contains 6
overlapping success runs of length 3.
4. Aki and Hirano [17] studied the distribution of success
runs of exact length k . In the above example number
of success runs of exact length 3 is 0.
5. Philippou[18] obtained the distribution of the number
of trials until the first occurrence of consecutive k
successes in Bernoulli trials with success probability as
the geometric distribution of order ( ) ( ) , ;
k
k G xp
In this paper, we have suggested alternative formulas for
Binomial distribution of order k based on success runs of based
on atleast length k by using Balls-into-cells technique with
direct sampling scheme with replacement. The same result
was extended by using inverse sampling scheme to obtain
alternative formula for Geometric distribution of order k and
negative Bionomial distribution of order
k
. Finally, using Polya
Eggenberger sampling scheme we have obtained alternative
formulas for Polya-Eggenberger distributions of order k and
Inverse Polya-Eggenberger distributions of order k [19,20].
Lemma: The number ways of distributing
r
indistinguishable balls in n cells such that each cell has atmost
( 1) k − balls is given as:
( ) ( )
( )
0
1
; 1
1
r
i
k k
i
n r ik n
F nr
i n
=
+ + −
= −
−
∑ (Riordan 1958) (1)
Binomial distribution of order k
Let
k
n
X the number of success runs of length at least
k
and
r be the number of success in n Bernoulli trials and n r − be
the number of failures.
Then,
Theorem 1:
( ) ( )
0
1
1
r xk
a j
k k
n
r xk j
n r x n r
PX x
x j
−
= =
− + − +
= = −
∑ ∑
Abstract
In this research paper, an attempt has been made to obtain alternative formulas for distributions of order k based on runs of at least length
k. First by using binomial scheme and ‘balls-into-cells’ technique an alternative formula for distribution of binomial distribution of order k as
defined by Goldstein [1]. Inverse Bionomial scheme was then used with ‘ball-into-cells technique to obtain alternate distribution of Geometric
distribution of order k and negative binomial distribution of order k. The results were further extended to obtain Polya-Eggenberger distribution
of order k and Inverse Polya-Eggenberger distributions of order k. All results were verified for exactness of probability.
Keywords : Polya-Eggenberger sampling scheme; Distributions of order k; ‘Balls-into-cells’ technique
Some Identities based on Success Runs of at
Least Length k
Biostat Biometrics Open Acc J 4(2): BBOAJ.MS.ID.555634 (2018)
Review Article
Volume 4 Issue 3 - January 2018
DOI: 10.19080/BBOAJ.2018.04.555634
Biostat Biometrics Open Acc J
Copyright © All rights are reserved by Sonali Bhattacharya
Biostatistics and Biometrics
Open Access Journal
ISSN: 2573-2633