Simple Expressions for Success Run Distributions in Bernoulli Trials

Transcription

1 Siple Epressions for Success Run Distributions in Bernoulli Trials Marco Muselli Istituto per i Circuiti Elettronici Consiglio Nazionale delle Ricerche via De Marini, Genova, Ital Eail: uselli@ice.ge.cnr.it Abstract New siple forulae for soe probabilit distributions of success runs in Bernoulli trials are found b using the classical definition of run. These epressions contain onl one suation of ordinar binoial coefficients and thus allow a faster and efficient coputation. Kewords: Bernoulli trials, nuber of success runs, longest success run, discrete distributions of order k. 1 Introduction Most recent studies on success runs in Bernoulli trials follow the fraework contained in the fundaental book of Feller 1968 and in particular his definition of run as a recurrent pattern. According to this definition two consecutive success runs a not be separated b an failure. As an eaple, the sequence SSSSSS where the sbol S denotes a success can be interpreted as containing 3 success runs of length 2 or 2 success runs of length 3. In practice, if we search for runs of length k, the counting of consecutive successes ust be restarted when the desired value k is reached see Feller, 1968, pag It follows fro this definition that the location of success runs in a sequence of n Bernoulli trials depends on the reference length k. Although this can see quite unnatural, soe atheatical derivations are greatl siplified particularl when dealing with asptotical epressions. Moreover, in soe cases, such as the probabilit distribution for the longest run, the final relation does not depend on the definition eploed during the proof. In the present work the stud of success runs in Bernoulli trials is carried out b using the classical definition which asserts that two consecutive runs ust be separated b one or ore failures. Following this approach in section 2 basic epressions for k and L n k 1 are derived, where k is the nuber of success runs with length k or ore and L n is the length of the longest success run in n Bernoulli trials. Unlike corresponding forulae obtained b hilippou and Makri 1986 and Hirano 1986, onl ordinar first order binoial coefficients are eploed and suations over an inde set deterined b the solutions of a diophantine equation are not involved. In particular, the epression for the distribution of L n reported in Burr and Cane 1961 and Godbole 1990 is again obtained b following a new procedure which allows to find sipler forulae containing a single suation section 3. Such an approach can be etended to the derivation of probabilit distributions of siilar rando variables, such as the kth order negative binoial and the kth order geoetric ones introduced b hilippou, Georghiou and hilippou, Their interest fro a coputational point of view is evident. 1

2 2 Basic epressions for the distribution of M k n and L n Referring to the classical definition of success runs, let S n, k and L n denote respectivel the nuber of successes, the nuber of success runs with length k or ore and the length of the longest success run in n Bernoulli trials, each with success probabilit p 0 p 1. The probabilit of having a failure will be denoted with q 1 p in the following. Let us begin with a theore that provides a first epression for the distribution of k : Theore 1 If k have k is the nuber of success runs with length k or ore in n Bernoulli trials, we where k, n and are positive integers. roof. Consider the following events 1 n k n k A j {A sequence of k consecutive successes starts in X j } p n q 1 where X j is the outcoe of the jth trial and denote with J {j 1,..., j } a subset of {1,..., n} containing eactl different indices; we can write k A j1 A j2 A j j 1,j 2,...,j j J A j having denoted with A j the copleent of the set A j. Thus b appling the inclusion-eclusion principle see Feller, 1968, pag. 106 we obtain k 1 r 2 where r is given b r A j1 A j j 1,...,j n k 1 j 1,...,j A j1 A j, S n n 3 The bounds for the nuber of failures can be easil obtained b noting that at least 1 failures are needed for separating the success runs with length k or ore starting in the positions j 1,..., j. On the other hand, the realization of these runs requires at least k successes. Now, suppose without loss of generalit that the indices j 1,..., j are ordered in an increasing wa j 1 j ; according to the classical definition of run the sequences of n trials contained in the event A j for j > 1 ust have a failure as the j 1th outcoe X j 1 F. It follows that the probabilit A j1 A j, S n n is nonnull onl if j 1 + k + 1 j 2, j 1 + k + 1 j, j + k 1 n Since j 1 1, b cobining these inequalities we obtain that r 0 for 1 + k k 1 > n > n + 1 k + 1 > n + 1 k having denoted with the integer not greater than. 2

3 In the opposite case we note that j 1,...,j A j1 A j, S n n N, p n q 5 where N, is the nuber of different sequences of n Bernoulli trials having eactl n successes and containing success runs with length k or ore. In fact, onl these sequences, each of which has probabilit p n q of occurring, provide a nonnull contribution to the suation on the left hand side in 5. A careful cobinatorial reasoning leads to an eplicit epression for N, ; in fact, if we consider + 1 the position of the failures we have different was of placing success runs of length k so that each of the is separated fro the neighbors at least b a failure. Then we can put the n k reaining n k successes into ever configuration in possible was. Thus we obtain for N, the following epression N, + 1 n k B considering 6 and 5 the equation 3 for r becoes r n k and 2 gives the desired epression 1 for k b 4. n k B interchanging the order of suation in 1 we have: p n q 6 if we use the upper bound for provided k n k 1 in+1, n p n q k n k 7 siilar to the epression for N n k found in Godbole 1990 b eploing the alternative definition of success run. The analog between the two forulae is ephasized b setting j in 7. Fro theore 1 we can directl obtain the relation for the distribution of the longest success run L n in n Bernoulli trials. For this ai it is useful to enunciate the following Lea 1 If k and n are positive integers, we have n k n k 0 for 0 < n/k roof. Consider the function f given b f 1 1 k +1 1 n k n+ 1 3

4 and copute its th derivative in the point 0 +1 f n k + i! n k 0 i1 0 Now, the direct coputation of the first derivatives ields epressions containing a coon ultiplicative factor 1 1 ν where ν is a positive integer. Consequentl we obtain n k 1! f Thus, consider the following two cases: when 0 < n + 1/k + 1 we have + 1 < n /k and for + 1 < n /k 8 consequentl n k n k n k 0 9 when n + 1/k + 1 < n/k we have n k 0 for ever + 1; then n k 0 for n /k < + 1 and 9 is again verified. B taking into account 8 we can write in+1, n k n k 0 n k n k and b virtue of lea 1 we obtain that the left hand side is null for 0 < n/k. This result allows to find the correct epression for the distribution of L n Theore 2 If L n denotes the length of the longest success run in n Bernoulli trials, we have n n L n k 1 p n q k n k n k 0 where k and n are positive integers. roof. Since M k n denotes the nuber of success runs with length k or ore, it follows fro 1 that L n k 1 n+1 1 k 1 4 n k n k p n q 10 11

5 and b interchanging the suations on and : L n k n k n k n k n k p n q p n q being Now, we note that L n k 1 1 L n k n k n k p n q and b interchanging the order of suation n in+1, L n k 1 p n q n k n k 12 In fact the inequalit 1 gives the upper bound + 1 for while n k leads to n /k. But, b virtue of 10 and lea 1 we obtain fro 12 the desired relation 11. Theore 2 provides the well known epression for L n k 1 alread obtained b Burr and Cane 1961 and Godbole 1990 with other ethods. Fro this result also the forulae for L n k, S n r and L n k, S n r presented in Gibbons 1971 follows directl. Incidentall, equation 12 could be obtained b setting 0 in 7; in this wa the achieveent of 11 would have been shorter. Unfortunatel, the proof of theore 1 onl holds for positive values of and thus the passage above would not be theoreticall acceptable. 3 Siplified epressions for soe success run distributions Fro equations 1 and 11 obtained for the distributions of k and L n respectivel follow soe interesting siplified epressions. The contain onl a single suation of ordinar first order binoial coefficients and therefore their corresponding coputation tie is considerabl lowered. Theore 3 If k have k is the nuber of success runs with length k or ore in n Bernoulli trials, we 1 where k, n and are positive integers. p k q 1 n k 1 n k 13 5

6 roof. B setting j n k in 1, we obtain k 1 1 n k +1 j0 n k j + 1 n k +1 p k q n k j0 n k j n k j + 1 Now, if we ake use of the ascal triangle identit, we have n k +1 j0 n k +1 j0 n k +1 j0 n k 1 p k+j q n k j n k j p/q j 14 n k j + 1 n k p/q j j n k j n k j n k + p/q j 1 j n k j n k j n k + p/q j n k + 1 j n k j j 1/q n k +1 n k + 1/q n k 15 In the last passage the following relation has been eploed see Feller, 1968, pag. 63: ν 0 h ν h ν r ν t ν h r 1 + t r which holds for r, h non-negative integers and for ever real nuber t. B substituting 15 in 14 we obtain the desired relation 13. Fro 13 it is possible to obtain the corresponding siplified epression for the distribution of the longest success run L n. This forula has alread been found b Labiris and apastavridis 1985 and Hwang 1986 in the stud of reliabilit for consecutive-k-out-of-n sstes. Corollar 1 If L n denotes the length of the longest success run in n Bernoulli trials, we have L n k p k q 1 n k 1 n k 16 where k and n are positive integers. roof. It is sufficient to proceed as in the first part of the proof of theore 2 b noting that L n k 1 1 L n k k p k q 1 n k 1 n k 6

7 The siplified forulae 13 and 16 per M k n and L n k 1 can be used for obtaining epressions with single suation of other interesting probabilit distributions. As an eaple let us consider the kth order negative binoial rando variable NB k,r defined as the waiting tie till the rth success run of length k or ore introduced b hilippou, Georghiou and hilippou, 1983, with the alternative definition of run. In case of classical definition of success run we have the following Theore 4 The rando variable NB k,r is characterized b the following probabilit distribution NB k,r +1 k+1 r 1 r 1 r 1 p k q 1 k 1 2 k roof. B definition of NB k,r ever sequence of n Bernoulli trials belonging to the event {NB k,r } ust end with k successes preceded b a failure. Thus, we have NB k,r p k q M k k 1 r 1 and b using the epression 13 for M k k 1 r 1 we obtain k NB k,r p k q 1 r+1 p k q 1 r 1 r 1 + 1k 1 + 1k k+1 r 1 r 1 r 1 p k q 1 k 1 2 k 1 1 This theore also allows to obtain the forula for the probabilit distribution of the kth order geoetric rando variable G k ; it is sufficient to set r 1 in 17 G k +1 k p k q 1 k 1 2 k 1 1 In this case the two epressions deriving fro different definitions of success run coincide Godbole, Acknowledgeent Thanks are due to prof. F. Fagnola for his valuable coents as well as to the referee for bringing to attention the paper of Labiris and apastavridis and the work of Hwang. References Burr, E.J. and G. Cane 1961, Longest run of consecutive observations having a specified attribute, Bioetrika 48, Feller, W. 1968, An Introduction to robabilit Theor and Its Applications, vol. 1 Wile, New York, 3rd ed. 7

8 Gibbons, J.D Nonparaetric Statistical Inference Mc Graw-Hill, New York. Godbole, A , Specific forulae for soe success run distributions, Statist. robab. Lett. 10, Hirano, K. 1986, Soe properties of the distributions of order k, in: A.N. hilippou, A.F. Horada and G.E. Bergu, eds., Fibonacci Nubers with Applications. roc. 1st Internat. Conf. on Fibonacci Nubers and their Applications Reidel, Dordrecht. Hwang, F.K. 1986, Siplified reliabilities for consecutive-k-out-of-n sstes, SIAM J. Alg. Disc. Meth. 7, Labiris, M. and S. apastavridis 1985, Eact reliabilit forulas for linear & circular consecutive-k-out-of-n:f sstes, IEEE Trans. Reliabilit R-34, hilippou, A.N., Georghiou, C. and G.N. hilippou 1983, A generalized geoetric distribution and soe of its properties, Statist. robab. Lett. 1, hilippou, A.N. and F.S. Makri 1986, Successes, runs and longest runs, Statist. robab. Lett. 4,