CHARACTERIZATION OF MARKOV-BERNOULLI GEOMETRIC DISTRIBUTION RELATED TO RANDOM SUMS

Transcription

1 Journal of Mathematics and Statistics 0 (2): 86-9, 204 ISSN: Science ublications doi:03844/jmssp ublished Online 0 (2) 204 ( CHARACTERIZATION OF MARKOV-BERNOULLI GEOMETRIC DISTRIBUTION RELATED TO RANDOM SUMS Mohamed Gharib, 2 Mahmoud M Ramadan and Khaled A H Al-Ajmi Department of Mathematics, Faculty of Science, Ain Shams University, Abbasia, Cairo, Egypt 2 Department of Mareting Administration, Community College, Taif University, Taif, Saudi Arabia 2 Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt Received ; Revised ; Accepted ABSTRACT The Marov-Bernoulli geometric distribution is obtained when a generalization, as a Marov process, of the independent Bernoulli sequence of random variables is introduced by considering the success probability changes with respect to the Marov chain The resulting model is called the Marov- Bernoulli model and it has a wide variety of application fields In this study, some characterizations are given concerning the Marov-Bernoulli geometric distribution as the distribution of the summation index of independent randomly truncated non-negative integer valued random variables The achieved results generalize the corresponding characterizations concerning the usual geometric distribution Keywords: Marov-Bernoulli Geometric Distribution, Random Sum, Random Truncation, Characterization INTRODUCTION The Marov-Bernoulli Geometric (MBG) distribution has been obtained by Anis and Gharib (982) in a study of Marov-Bernoulli sequence of random variables (rv introduced by Edwards (960), who generalized the (usual) independent Bernoulli sequence of rv s by considering the success probability changes with respect to a Marov chain The resulting model is called the Marov-Bernoulli Model (MBM) or the Marov modulated Bernoulli process (Ozeici, 997) Many researchers have been studied the MBM from the various aspects of probability, statistics and their applications, in particular the classical problems related to the usual Bernoulli model (Anis and Gharib, 982; Gharib and Yehia, 987; Inal, 987; Yehia and Gharib, 993; Ozeici, 997; Ozeici and Soyer, 2003; Arvidsson and France, 2007; Omey et al, 2008; Maillart et al, 2008; acheco et al, 2009; Ceanavicius and Vellaisamy, 200; Minova and Omey, 20) Further, due to the fact that the MBM operates in a random environment depicted by a Marov chain so that the probability of success at each trial depends on the state of the application of stochastic processes and thus used by numerous authors in, stochastic modeling (Switzer, 967; 969; edler, 980; Satheesh et al, 2002; Özeici and Soyer, 2003; Xealai and anaretos, 2004; Arvidsson and France, 2007; Nan et al, 2008; acheco et al, 2009; Doubleday and Esunge, 20; ires and Diniz, 202) Let X, X 2, be a sequence of Marov-Bernoulli rv s with the following matrix of transition probabilities Equation (): ( ρ ) p ( ρ ) p ( ρ )( p) ρ + ( ρ ) environment, this model represents an interesting where, 0 p and 0 ρ Corresponding Author: Mahmoud M Ramadan, Department of Mareting Administration, Community College, Taif University Saudi Arabia 0 X i X i + 0 and initial distribution:, p ( ) ( 0) X p X () Science ublications 86

2 Mohamed Gharib et al / Journal of Mathematics and Statistics 0 (2): 86-9, 204 The sequence {X i } with the transition matrix () and the above initial distribution is called the MBM If E i, i 0, are the states of the Marov system given by (), then the parameter ρ which is usually called the persistence indicator of E 0, is the correlation coefficient between X i and X i+, i, 2, (Anis and Gharib, 982) If N is the number of transitions for the system defined by () to be in E for the first time then N has the MBG distribution given by Equation (2): α, 0 ( N ) p α ( ) t t,, (2) where: α -p and t ρ + (l ρ) α α +(l α) ρ The MBG distribution (2) will be denoted by MBG(α, ρ) and we shall write N MBG(α, ρ) Let U ( E(s U ) be the probability generating function (pgf) of an integer valued random variable U, s The pgf of N is given, using (2), by Equation (3): N ( α )( ρ (3) 0 ( t s p s Few authors have treated the characterization problem of the MBG distribution (Yehia and Gharib, 993; Minova and Omey, 20) In this study some new characterizations are given for the MBG distribution by considering it as the distribution of the summation index of a random sum of randomly truncated non-negative integer valued rv s The scheme of geometric random sum of randomly truncated rv s is important in reliability theory, especially in the problem of optimal total processing time with checpoints (Dimitrov et al, 99) The achieved results generalize those given in (Khalil et al, 99) 2 RANDOM SUM OF TRUNCATED RANDOM VARIABLES Consider a sequence {X n, n } of independent identically distributed (iid) non-negative integer valued rv s with probability mass function (pmf) 0 p (X ); 0,, 2, Let {Y n, n } be another sequence of iid nonnegative integer valued random variables, independent of {X n }, with pmf q (Y ); 0,, 2,, {Y n } is called the truncating process ut p (X Y) and assume 0< p < Let N inf { : X <Y } Clearly, q 0 (N- ) p (-p), 0,,2,, ie, N- has a geometric distribution with parameter p Remar If the sequence {X n, n } follows the MBM (), where the event {(X i Y i )} indicates the state E 0 and the event {(X i <Y i )} indicates the state E, with the initial distribution: X Y p and X < Y p Then the rv N (the smallest number of transitions for the system to be in state E for the first time) wouldhave the MBG distribution defined by (2) (Anis and Gharib, 982) Equation (2): Define: N + N (2) 0 Z Y X where, for convenience Y 0 0 The rv Z is the random sum of truncated rv s (Khalil et al, 99) It is nown that the pgf Z ( of Z is given by (Khalil et al, 99) Equation (22 to 25): Z s s / Q s, (22) where: X ( < ), (23) s E S I X Y p s q γ 0 γ + Y ( ) γ, (24) Q s E S I X Y q s p γ 0 and I(A) is the indicator function of the set A Corollary E Z ( p 0 q γ + γ ) ( q 0 p γ + γ ) The result is immediate since E(Z) z () Corollary 2 (25) If the truncating process {Y n } is such that Y n ~MBG(α, ρ), for α (0,) and ρ [0, ] Then the pgf of Z is given by Equation (26 and 27): Science ublications 87

3 Mohamed Gharib et al / Journal of Mathematics and Statistics 0 (2): 86-9, 204 Z ( t X ( t { t X ts } s α X ( t, ( α ) ( t) s αx ( t ( t ( α) (26) 0)< Then Z has the same distribution as X ( Z X) if and only if X has a MBG distribution roof Assume that X MBG(β, ρ ), for some β (0, ) and ρ [0, ],ie,: d and: X ( t ) ( t) ( t ) E Z Z, X where, t ρ+( ρ)α roof Since Y n ~MBG(α, ρ), then: α, 0 ( Yn ) q α ( t) t,, 2, Hence, using (23), we have Equation 28 and 29: γ 0 γ + s p s q 0 γ γ γ γ + X p q + p s q α ts, and, using (24), we have: 0 γ + γ 0 γ Q s q p q s p (27) (28) ( α ) α ( t) s pγ ( t (29) + ( α ) ( t γ { ρ α ( ρ ) X } s ts + γ Substituting the expressions of ( and Q ( given respectively by (28) and (29) into (22), we get (26) E(Z) is obtained readily from (25) or (26) 3 CHARACTERIZATIONS OF THE MBG DISTRIBUTION Consider the random sum given by (2) Theorem Let Y, have a MBG distribution with some parameters α (0, ) and ρ [0, ] and let X satisfy (X γ β, 0 p β ( ll ),, 2,, where, l ρ +( ρ )β Then the pgf of X is given by Equation (3 and 32): X Z ( β )( ρ (3) ( l Substituting, (3) into (26), we obtain: ( β )( t( ρts ) s lt lts ( t)( β )( ρts ) ( β )( t( ρts ) 2 ts β ρt ( β ) s β ρt ( β ) t ( β )( ρ2s ), ( D (32) where, ρ 2 ρ t and D β + ρ 2 ( β) It follows from (32), that the rv Z has a MBG distribution with parameters β (0, ) and ρ 2 [0, ] Conversely, assume that Z ( X ( where Y MBG(α, ρ) Hence, we have to show that X ( is the pgf of MBG distribution Replacing Z ( by X ( in (26), yields in: ( t X (ts[ s{ ( t) X (t}] X ( Or: ( t X (t ( X (+s( t) X ( X- (t Dividing both sides of this equation by ( ( t X ( X (t, we get: s( t) + ts ts s ts s s X X Now, using partial fractions for the middle term and using some manipulations, we can write Equation (33): ( ) ( ) X ts X s ts s (33) Science ublications 88

4 Mohamed Gharib et al / Journal of Mathematics and Statistics 0 (2): 86-9, 204 X Or, H(t H(, where, H ( utting s, we get: s ( X H t H lim E X C, say ( ) Therefore, the solution of (33) is given by: X ( ) s + C s Choosing C β/( β){ [α+( α)ρ]ρ }and recalling that α + ( α)ρ t, we finally get: X ( β )( ρ ( l where, l ρ +( ρ )β Which is the pgf of the MBG(β, ρ ) distribution Hence X MBG(β, ρ ) This completes the proof of Theorem Remar 2 It follows from Theorem that when X has the MBG distribution then the random sum Z and the summands have distributions of the same type and in this case the summands are called N-sum stable (Satheesh et al, 2002) This result is valid, also, as a consequence of the fact that geometric random sums are stable in the same sense Another characterization for the MBG distribution can be obtained in terms of the expected values of Z and X Theorem 2 Let Y have a MBG distribution with parameters α (0,) and ρ [0,] and consider A {(α, ρ): α (0, ), ρ [0, ]} Then X has a MBG(β, ρ ) distribution if and only if (α,ρ) A, E(Z) β/ ( β)( ρ 2 )], where ρ 2 ρ t and t ρ + ( ρ)α roof If X MBG(β, ρ ) for some (β, ρ ) A, then it follows from Theorem that Z MBG(â, ρ 2 ), where ρ 2 ρ t and t ρ + ( ρ)α Consequently, E(Z) β/[( β)( ρ 2 )], Conversely, suppose that: E(Z) β/[( β)( ρ 2 )] for some(β, ρ ) A Then from (27), we have: β ( β )( ρ ) 2 X ( t ) ( t) ( t) Solving this equation with respect to X (, one has: X ( β )( ρt ) ( lt) X ( α β ) t,, A Which is the pgf of the MBG (β, ρ ) distribution Hence X MBG(β, ρ ) This completes the proof of Theorem 2 The following theorem expresses the relation between the distribution of Z and the distribution of the truncating process {Y n } Theorem 3 Let X have a MBG(β,ρ ) distribution for some parameters β (0,) and ρ [0,] and let Y satisfy q 0 d (Y 0) < Then Z X if and only if Y follows a MBG distribution roof Since X MBG(β, ρ ), for some β (0, ) and ρ [0, ], then: β, 0 p β ( ll ), 2,, where, l ρ +( ρ )β roceeding as in Corollary 2, one has: γ 0 γ + s p s q and: ( β )( q0 ) β ( l) s qγ ( l + ( β ) ( γ γ 2 [ l ρq0 lsρ q0 β ρ Y ls ], l l (34) Science ublications 89

5 Mohamed Gharib et al / Journal of Mathematics and Statistics 0 (2): 86-9, 204 γ 0 γ Q s p s p β q + q ( l β ( l + q ρ ( β) 0 Y 0 Y l l Substituting (34) and (35) into (22), we obtain: l l l Z ls l q0 Y ls 0 + ( Y ) (35) ( β ) ρq 0 ρ ( q0 ) s β ( ρ ) Y ρ ( β ) β (36) ( β ) ρ + ( ρ ) β ρq 0 lρ ( q0 ) s β ( ρ ) Y ( l ( l ρ ( β )( q ) β ( l d Now assume that Z X, then Z ( X ( Consequently, equating (3) and (36) and then solving for ( l, we get: Y Y ( l ( ) ( βq0 + ρ q0 β s β From which we finally have: Y { ( ρ ) ρ β } q0 q0 q0 s ( l Which is the pgf of a MBG(γ, ρ 2 ) distribution with, γ q 0 and ρ 2 (ρ q 0) [ ρ β ( q 0)] Hence Y MBG(γ, ρ 2 ) The only if part of the proof follows directly by applying Theorem This completes the proof of Theorem 3 Remar 3 The results of (Khalil et al, 99) follow as special cases from our corresponding results when the MBM () reduces to the independence case by putting the correlation parameter ρ 0 4 CONCLUSION In this study three characterizations for the Marov- Bernoulli geometric distribution are proved These results extend the corresponding characterizations of the geometric distribution Further, the achieved results have a direct relevance to the stability problem of random sums of random variables Moreover, the given characterization theorems will be useful, in understanding the missing lin between the mathematical structure of Marov-Bernoulli geometric distribution and the actual behavior of some real world random phenomena 5 REFERENCES Anis, AA and M Gharib, 982 On the Bernoulli Marov sequences roceedings of the 7th Annual Conference Stat Math Inst Stat Studies Res Dec 3-6, Cairo University ress, Cairo, Egypt, pp: -2 Arvidsson, H and S France, 2007 Dependence in nonlife insurance UUDM roject Report, 23 Department of Mathematics Ceanavicius, V and Vellaisamy, 200 Compound oisson and signed compound oisson approximations to the Marov binomial law Bernoulli, 6: 4-36 Dimitrov, B, Z Khalil, N Kolev and etrov, 99 On the optimal total processing time using checpoints IEEE Trans Software Eng, 7: DOI: 009/ Doubleday, K J and J N Esunge, 20 Application of Marov chains to stoc trends J Math Stat, 7: DOI: 03844/jmssp Edwards, AWF, 960 The meaning of binomial distribution Nature, 86: 074 DOI: 0038/86074a0 Gharib, M and AY Yehia, 987 A limit theorem for the sum of n-marov-bernoulli random variables Egypt Stat J, 3: 53-6 Inal, C, 987 On the limiting distribution for Bernoulli trials within a Marov-chain context Commun Fac Sci Univ An Ser A, 36: Khalil, Z, B Dimitrov and J Dion, 99 A characterization of the geometric distribution related to random sums Commun Stat-Stochastic Models, 7: DOI: 0080/ Maillart, LM, CR Cassady and J Honeycutt, 2008 A binomial approximation of lot yield under Marov modulated Bernoulli item yield IIE Trans, 40: DOI: 0080/ Minova, LD and E Omey, 20 A new Marov binomial distribution HUB Res Econ Manage Nan, F, Y Wang and X Ma, 2008 Application of multiscale hidden Marov modelling Wavelet coefficients to fmri activation detection J Math Statist, 4: DOI: 03844/jmssp Science ublications 90

6 Mohamed Gharib et al / Journal of Mathematics and Statistics 0 (2): 86-9, 204 Omey, E, J Santos and SV Gulc, 2008 A Marovbinomial distribution Applicable Anal Discrete Math, 2: DOI: 02298/AADM080038O Ozeici, S and R Soyer, 2003 Bayesian analysis of Marov modulated Bernoulli processes Math Methods Operat Res, 57: DOI: 0007/s Ozeici, S, 997 Marov modulated Bernoulli process Math Methods Operat Res, 45: DOI: 0007/BF acheco, A, LC Tang, NU rabhu, 2009 Marov- Modulated rocesses and Semiregenerative henomena st Edn, World Scientific ub Co Inc, UK ISBN-0: , pp: 224 edler, J, 980 Effect of dependence on the occupation time in a two-state stationary Marov chain J Am Statist Associat, 75: DOI: 0080/ ires, RM and CAR Diniz, 202 Correlated binomial regression models Comput Statist Data Analysis, 56: DOI: 006/jcsda Satheesh, S, NU Nair and E Sandhya, 2002 Stability of random sums Stochastic Modelling Applic, 5: 7-26 Switzer,, 967 Reconstructing patterns from sample data Ann Math Statist, 38: DOI: 024/aoms/ Switzer,, 969 Mapping a geographically correlated environment st Edn, Defense Technical Information Center, pp: 38 Xealai, E and J anaretos, 2004 A binomial distribution with dependent trials and its use in stochastic model evaluation Commun Stat Theory Meth, 33: Yehia, AY and M Gharib, 993 Characterizations of the Marov-Bernoulli geometric distribution Micro Reliab, 33: Science ublications 9