الفهرس 
Basic Definitions, Properties andOnly 14 pages are availabe for public view
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Abstract
The uses of the binomial and multinomial distributions in statistical modelling and
analyzing discrete data are very well understood, with a huge variety of applications and
appropriate software, but there are plenty of real-life examples where these simple models
are inadequate. Therefore, it seems wise to consider flexible alternative models to take into
account the overdispersion or underdispersion (Hinde & Demetrio (1998)). Thus, the binomial
and Poisson distributions have been generalized in several ways to handle the problem of
dispersion inherent in the analysis of discrete data that may arise with the presence of
aggregation of the individuals. The binomial distribution has been generalized in various ways.
Rudolfer (1990), Madsen (1993) and Luceno & Ceballos (1995) have summarized most of these
generalizations. Among these extensions, there are the generalized binomial distribution
introduced by Edwards (1960) and the multiplicative and the additive generalized binomial
distributions which were derived by Altham (1978).
As finite Markovian models are extensively used in varies application fields, the
generalized Markov-binomial model (Markov- Bernoulli model MBM, also called Markov
modulated Bernoulli process ( Ozekici(1997))) introduced by Edwards (1960) have been
studied by many researchers from the various aspects of probability, statistics and their
applications, in particular the classical problems related to the usual Bernoulli model (Anis &
Gharib (1982), Arvidsson & Francke (2007), Cmey et al (2008) , Cekanavicius & Vellaisamy
(2010) , Gharib & Yehia(1987), Inal(1987), Maillart et al.(2008), Minkova & Omey (2011),
Ozekici (1997), Ozekici et al (2003),Pacheco et al. (2009), Yehia & Gharib(1993) and others.).
Further, due to the fact that the MBM operates in a random environment depicted by a
Markov chain so that the probability of success at each trial depends on the state of the
environment, this model represents an interesting application of stochastic processes, and
thus used by numerous authors in stochastic modelling (see for example, Switzer (1967, 1971),
Pedler (1980), Xekalaki & Panaretos (2004), Arvidsson, & Francke (2007), Pires & Diniz (2012)).
The present thesis is devoted to study the probability distributions related to the
Markov- Bernoulli sequence of random variables (the MBM) from some aspects such as
distributional properties, characterizations, limit theorems, generalizations, and throw the
light on some applications.
The thesis consists of five chapters and an introduction. The introduction is devoted to
show the actuality of the subject of study and to give a historical survey about it. Chapter one
is devoted to give the basic definitions, properties and preliminary results concerning the
Markov- Bernoulli sequence of random variables (MBM). Chapter one, also, throw light on
some generalizations of MBM and some examples of its applications.
Chapter two is concerned with the Markov binomial and Markov negative binomial
distributions, exploring their properties, characteristic functions and relations to other
distributions. This chapter contains, also, a numerical study to specify the descriptive
characteristics of the Markov binomial distribution, besides a detailed investigation for the
generalized Markov-binomial distribution introduced by Xekalaki & Panaretos (2004).
Chapter three is devoted to investigate the properties of the Markov-Bernoulli
geometric (MBG) distribution through characterizing it. It is worth mentioning that the results
of both sections 3.2 and 3.3 are totally new and is published respectively, in Journal of
Mathematics and Statistics (Vol. 10, No. 2, 186-191, 2014), and in International Journal of
Statistics and Probability (Vol. 3, No. 3, 138-146, 2014).
Chapter four is devoted to investigating the limiting behavior of the sum of n- MarkovBernoulli random variables. In section 4.1 a new prove is given for the central limit theorem
using generating functions technique. In section 4.2 we discuss in details the results of Gharib
et al. (1987) concerning uniform estimates of the rate of convergence in the central limit
theorem. Section 4.3 is devoted to discussing the results of Gharib et al. (1991) concerning
limit theorem in the space 𝐿𝜋, (1 ≤ 𝜋 ≤ ∞).
In chapter five, a new method is introduced for adding two parameters to an existing
distribution. This new technique extends the methods of Edwards (1960) and Marshall and
Olkin (1997) for adding a parameter to a family of distributions. The method is of direct
relevance to the Markov-Bernoulli geometric distribution and is applied in particular, to a one
parameter Burr XII distribution to yield a three parameter extended Burr XII distribution which
may serve as a competitor to such commonly used three parameters families of distributions.
The results of this chapter are totally new and are submitted for publication in an international
specialized journal.