Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials
ABSTRACT
In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (X0n,k10,X1n,k11;Y0n,k20,Y1n,k21) where for i=0,1,Xin,k1i denotes the number of occurrences of i-runs of length k1i in the first component and Yin,k2i denotes the number of occurrences of i-runs of length k2i in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ1 and Λ2 of lengths k1 and k2 respectively and obtain the pgf of joint distribution of (Xn,Λ 1,Yn,Λ2 ) using method of conditional probability generating functions where Xn,Λ1(Yn,Λ2) denotes the number of occurrences of pattern Λ1(Λ2 ) of length k1 (k2) in the first (second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.
In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (X0n,k10,X1n,k11;Y0n,k20,Y1n,k21) where for i=0,1,Xin,k1i denotes the number of occurrences of i-runs of length k1i in the first component and Yin,k2i denotes the number of occurrences of i-runs of length k2i in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ1 and Λ2 of lengths k1 and k2 respectively and obtain the pgf of joint distribution of (Xn,Λ 1,Yn,Λ2 ) using method of conditional probability generating functions where Xn,Λ1(Yn,Λ2) denotes the number of occurrences of pattern Λ1(Λ2 ) of length k1 (k2) in the first (second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.
KEYWORDS
Markov Dependent Bivariate trials, Conditional Probability Generating Function, Joint Distribution
Markov Dependent Bivariate trials, Conditional Probability Generating Function, Joint Distribution
Cite this paper
nullK. Kamalja and R. Shinde, "Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials," Open Journal of Statistics, Vol. 1 No. 2, 2011, pp. 115-127. doi: 10.4236/ojs.2011.12014.
nullK. Kamalja and R. Shinde, "Runs and Patterns in a Sequence of Markov Dependent Bivariate Trials," Open Journal of Statistics, Vol. 1 No. 2, 2011, pp. 115-127. doi: 10.4236/ojs.2011.12014.
References
[1] S. J. Schwager, “Run Probabilities in Sequences of Markov-Dependent Trials,” Journal of American Statistical Association, Vol. 78, No. 381, 1983, pp. 168-175. doi:10.2307/2287125 [2] N. Balakrishnan and P. S. Chan, “Start-up Demonstration Tests with Rejection of Units upon Observing d Failures,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 1, 2000, pp. 184-196. doi:10.1023/A:1004101402897 [3] W. Feller, “An Introduction to Probability Theory and Its Applications,” 3rd Edition, John Wiley & Sons, Hoboken, 1968. [4] K. D. Ling, “On Binomial Distributions of Order k,” Statistics & Probability Letters, Vol. 6, No. 4, 1988, pp. 247-250. doi:10.1016/0167-7152(88)90069-7 [5] S. Aki and K. Hirano, “Number of Success-Runs of Specified Length until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 4, 2000, pp. 767-777. doi:10.1023/A:1017585512412 [6] A. M. Mood, “The Distribution Theory of Runs,” Annals of Mathematical Statistics, Vol. 11, No. 4, 1940, pp. 367-392. doi:10.1214/aoms/1177731825 [7] S. Aki and K. Hirano, “Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials,” Annals of Institute of Statistical Mathematics, Vol. 51, No. 1, 1999, pp. 17-29. doi:10.1023/A:1003874900507 [8] M. Uchida, “On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables,” Annals of Institute of Statistical Mathematics, Vol. 50, No. 4, 1998, pp. 655-671. doi:10.1023/A:1003756712643 [9] D. L. Antzoulakos, “Waiting Times for Patterns in a Sequence of Multistate Trials,” Journal of Applied Probability, Vol. 38, No. 2, 2001, pp. 508-518. doi:10.1239/jap/996986759 [10] K. Inoue and S. Aki, “Generalized Waiting Time Problems Associated with Pattern in Polya’s Urn Scheme,” Annals of Institute of Statistical Mathematics, Vol. 54, No. 3, 2002, pp. 681-688. doi:10.1023/A:1022431631697 [11] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Patterns in the Sequence of Markov Dependent Multi- State Trials,” 2010 Submitted. [12] S. Aki and K. Hirano, “Waiting Time Problems for a Two-Dimensional Pattern,” Annals of Institute of Statistical Mathematics, Vol. 56, No. 1, 2004, 169-182. doi:10.1007/BF02530530 [13] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Runs in a Sequence of Higher-Order Two-State Markov trials,” Annals of Institute of Statistical Mathematics, Vol. 58, No. 1, 2006, pp. 537-554. doi:10.1007/s10463-005-0024-6 [14] A. A. Salvia and W. C. Lasher, “2-Dimensional Consecutive k-out-of-n: F Models,” IEEE Transactions on Reliability, Vol. R-39, No. 3, 1990, pp. 382-385. doi:10.1109/24.103023 [15] R. L. Shinde and K. S. Kotwal, “On the Joint Distribution of Runs in the Sequence of Markov Dependent Multi- State Trials,” Statistics & Probability Letters, Vol. 76, No. 10, 2006, pp. 1065-1074. doi:10.1016/j.spl.2005.12.005
[1] S. J. Schwager, “Run Probabilities in Sequences of Markov-Dependent Trials,” Journal of American Statistical Association, Vol. 78, No. 381, 1983, pp. 168-175. doi:10.2307/2287125 [2] N. Balakrishnan and P. S. Chan, “Start-up Demonstration Tests with Rejection of Units upon Observing d Failures,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 1, 2000, pp. 184-196. doi:10.1023/A:1004101402897 [3] W. Feller, “An Introduction to Probability Theory and Its Applications,” 3rd Edition, John Wiley & Sons, Hoboken, 1968. [4] K. D. Ling, “On Binomial Distributions of Order k,” Statistics & Probability Letters, Vol. 6, No. 4, 1988, pp. 247-250. doi:10.1016/0167-7152(88)90069-7 [5] S. Aki and K. Hirano, “Number of Success-Runs of Specified Length until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k,” Annals of Institute of Statistical Mathematics, Vol. 52, No. 4, 2000, pp. 767-777. doi:10.1023/A:1017585512412 [6] A. M. Mood, “The Distribution Theory of Runs,” Annals of Mathematical Statistics, Vol. 11, No. 4, 1940, pp. 367-392. doi:10.1214/aoms/1177731825 [7] S. Aki and K. Hirano, “Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials,” Annals of Institute of Statistical Mathematics, Vol. 51, No. 1, 1999, pp. 17-29. doi:10.1023/A:1003874900507 [8] M. Uchida, “On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables,” Annals of Institute of Statistical Mathematics, Vol. 50, No. 4, 1998, pp. 655-671. doi:10.1023/A:1003756712643 [9] D. L. Antzoulakos, “Waiting Times for Patterns in a Sequence of Multistate Trials,” Journal of Applied Probability, Vol. 38, No. 2, 2001, pp. 508-518. doi:10.1239/jap/996986759 [10] K. Inoue and S. Aki, “Generalized Waiting Time Problems Associated with Pattern in Polya’s Urn Scheme,” Annals of Institute of Statistical Mathematics, Vol. 54, No. 3, 2002, pp. 681-688. doi:10.1023/A:1022431631697 [11] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Patterns in the Sequence of Markov Dependent Multi- State Trials,” 2010 Submitted. [12] S. Aki and K. Hirano, “Waiting Time Problems for a Two-Dimensional Pattern,” Annals of Institute of Statistical Mathematics, Vol. 56, No. 1, 2004, 169-182. doi:10.1007/BF02530530 [13] K. S. Kotwal and R. L. Shinde, “Joint Distributions of Runs in a Sequence of Higher-Order Two-State Markov trials,” Annals of Institute of Statistical Mathematics, Vol. 58, No. 1, 2006, pp. 537-554. doi:10.1007/s10463-005-0024-6 [14] A. A. Salvia and W. C. Lasher, “2-Dimensional Consecutive k-out-of-n: F Models,” IEEE Transactions on Reliability, Vol. R-39, No. 3, 1990, pp. 382-385. doi:10.1109/24.103023 [15] R. L. Shinde and K. S. Kotwal, “On the Joint Distribution of Runs in the Sequence of Markov Dependent Multi- State Trials,” Statistics & Probability Letters, Vol. 76, No. 10, 2006, pp. 1065-1074. doi:10.1016/j.spl.2005.12.005