Some Identities based on Success Runs of at Least Length k

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:


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][9][10][11][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. [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. [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

Philippou
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:

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, Where, First consider x runs of exact length k successes each. x successes and n-r failures can be arranged in r xk − remaining successes can be distributed into ( 1) Thus, leading to the result. This is an alternative representation of probability distribution function of  Then the remaining number of cells formed by n r − failures and x successes will be 1 n r x − + + each assumed to be having exactly ( 1) k − successes.
So, ( )     10  9  2 8  2 8  3 7  3 7   10  10  10  9  64  1  2        The number of ways of distributing r k − successes preceding failures n r − such that not more than 1 k − 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

Some Examples:
Hence, the result. This is thus an alternative formula for Geometric distribution of order k given by Philippou [18]. Where, ( 1) xk n k a k

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 1 k − successes which is given by (1) replacing r by r xk − and n by as Hence, the probability. Further, This is thus an alternative formula for Negative Binomial distribution of order given by Philippou [18].

Some examples
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].      Balls are drawn one-by-one with s replacement along with additional balls of the same colour of ball drawn. The process is continued till n balls have been drawn [21].