Some Identities based on Success Runs of at Least Length k
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: sonali_bhattacharya@scmhrd.edu
How to cite this article: Sonali B. Some Identities based on Success Runs of at Least Length k . Biostat Biometrics Open Acc J. 2018; 4(2): 555634. DOI: 10.19080/BBOAJ.2018.04.555634
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
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 ssuccess 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, (Gk ( x; p ))
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 (k-1) balls is given as:
Binomial distribution of order k
Let Xkn 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.
Proof: First consider x runs of exact length k successes each. x successes and n-r failures can be arranged in ways. r - xk remaining successes can be distributed into n-r +1 cells such that no cells have more than k -1 successes in
Thus, leading to the result. This is an alternative representation of probability distribution function of
For the maximum possible value of r,
Let us assume that there are x successes of exact length k + (k -1) followed by a failure each.
Then the remaining number of cells formed by n - r failures and X successes will be n - r + x +1 each assumed to be having exactly (k-1) successes.
Geometric distribution of order k
Let the first success run of length k occurs atN thk trial. Then,
Theorem 2
The number of ways of distributing r — k successes preceding failures n — r such that not more than k — 1 successes occur proceeding the n — r failures then by using (1) and replacing n by n — r and r by r — k is given by
Hence, the result. This is thus an alternative formula for Geometric distribution of order k given by Philippou [18].
Remark 1: For k -1 (3) gives the probability mass function of Geometric Distribution.
Proof
Let there be r successes and n — r failures. r — xk successes are to be distributed into n — r cells preceding the n — r failures such that no cell receives more than k — 1 successes which is given by (1) replacing r by r — xk and n by as
This is thus an alternative formula for Negative Binomial distribution of order given by Philippou [18].
Some examples
For k = 1 → a = x and r = x
Polya-eggenberger distribution of order k
Let us assume an urn contains α white and b black balls. A ball is drawn, its colour is noted and it is returned to the urn with S1 additional balls of the same colour. The process is continued till n balls have been drawn.
Let Xsn;k be the number of white ball runs of at least length k in n trials. Then,
The proof follows from theorem 1. This is thus an alternative formula for Polya distribution of order given by Charalambides [18] Sen et al. [19].
Inverse polya-eggenberger distribution of order k
Let us assume an urn contains a white and b black balls. Balls are drawn one-by-one withs replacement along with additional balls of the same colour of ball drawn. The process is continued till n balls have been drawn [21].
Let Nx,sk be the number of trials required for xth run of k white balls to occur (Nx,0k =Nxk). Then,
Then, using theorem 3 we have,
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