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Extension of Generalized Bernoulli Learning Models

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ABSTRACT
In this article, we study the generalized Bernoulli learning model based on the probability of success pi = ai /n where i = 1,2,...n 0<a1<a2<...<an<n and n is positive integer. This gives the previous results given by Abdulnasser and Khidr [1], Rashad [2] and EL-Desouky and Mahfouz [3] as special cases, where pi = i/n pi = i2/n2 and pi = ip/np respectively. The probability function P(Wn = k) of this model is derived, some properties of the model are obtained and the limiting distribution of the model is given.
 
Cite this paper
El-Desouky, B. , Shiha, F. and Magar, A. (2015) Extension of Generalized Bernoulli Learning Models. Open Journal of Modelling and Simulation, 3, 26-31. doi: 10.4236/ojmsi.2015.31003.
 
References
[1]Abdulnasser, T. and Khidr, A.M. (1981) On the Distribution of a Sum of Non-Identical Independent Binomials. Journal of University of Kuwait, 8, 109-115.
 
[2]Rashad, A.M. (1998) On the Sum of Non-Identical Random Variables. M.Sc. Thesis, Faculty of Science, Aswan.
 
[3]EL-Desouky, B.S. and Mahfouz, K.M. (2012) A Generalization of Bernolli Learning Models. Kuwait Journal of Science and Engineering, 39, 31-44.
 
[4]Janardan, K.G. (1997) Bernoulli Learing Models: Uppulur numbers. In: Balakrishnan, N., Ed., Advances in Combinatorial Methods and Applications to Probability and Statistics, Birkhäuser. Statistics for Industry and Technology, Boston, 471-480.
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[5]Parzen, E. (1960) Modern Probability Theory and Its Applications. John Wiley & Sons, Hoboken.
 
[6]Comtet, M.L. (1972) Nombers de Stirling generoux et functions symetriques. C. R. Acad. Sci. Paris, Ser. A, 275, 747-750.
 
[7]Comtet, M.L. (1974) Advanced Combinatorics. Reidel, Dordrecht-Holland.
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[8]EL-Desouky, B.S. and Cakic, N.P. (2011) Generalized Higher Order Stirling Numbers. Mathematical and Computer Modelling, 54, 2848-2857. http://dx.doi.org/10.1016/j.mcm.2011.07.005
 
[9]EL-Desouky, B.S., Cakic, N.P. and Mansour, T. (2010) Modified Approach to Generalized Stirling Numbers via Differential Operators. Applied Mathematics Letters, 23, 115-120.
http://dx.doi.org/10.1016/j.aml.2009.08.018
 
[10]Riordan, J. (1968) Combinatorial Identities. John Wiley & Sons, New York. London, and Sydney.
 
[11]Beardon, A.F. (1996) Sums of Powers of Integers. The American Mathematical Monthly, 103, 201-213.
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[12]Sun, Y. (2006) Two Classes of p-Stirling Numbers. Discrete Mathematics, 306, 2801-2805.
http://dx.doi.org/10.1016/j.disc.2006.05.016                                                                           eww150115lx
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