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Papers by Mohamed I. Riffi

Research paper thumbnail of A New Method of Generating Life-Time Distributions

Abstract. Statisticians are interested in improving the ability of classical distributions to app... more Abstract. Statisticians are interested in improving the ability of classical distributions to appropriately fit real-world data and accurately describe its characteristics. Typically, this can be achieved by extending known distributions by incorporating extra parameters or compounding distributions. In this paper, a flexible general method to obtain new families of distributions is proposed. A number of known methods turn out to be special cases of the proposed method in this paper. Some examples are given to demonstrate the power of the proposed method such as the exponential and the Weibull distribution.

Research paper thumbnail of Simulating random walk using Wolfram Mathematica

Research paper thumbnail of Implementing the Moments of Gamma Spacings Using Wolfram Mathematica

Research paper thumbnail of The Cubic Rank Transmuted Gompertz- Makeham Distribution: Properties and L-moments

Research paper thumbnail of Mathematica code for constructing a 100(1-alpha)% confidence interval for the mean of the normal distribution when its variance is known

Research paper thumbnail of Short-Time Asymptotics of the Neumann Heat Kernel for Antipodal Points on the Exterior of a Ball

Research paper thumbnail of Simulating the minimum spacing of exponential order statistics

Research paper thumbnail of Bayesian Inference on Finite Mixtures of Poisson Distributions

Mixed Poisson distributions are widely used in various disciplines to model data in which each ob... more Mixed Poisson distributions are widely used in various disciplines to model data in which each observation is assumed to come from one of a number of Poisson distributions with different parameters. In this thesis, we investigate the Bayesian estimation for the finite Poisson mixture model using the Gibbs sampler as an important one of the MCMC methods. Our approach in this thesis depends on using the Gibbs sampler to simulate a Markov chain which has the posterior density as its stationary distribution. Then we use the resulting sample to make the suitable Bayesian computations and draw conclusion about the unknown parameters of the Poisson mixture model. We conclude this thesis by presenting a real data example to illustrate our methodology. vi Contents Abstract vi pdf probability density function. pmf probability mass function. xii algorithm is one of the most basic Markov Chain Monte Carlo Methods that is used in Bayesian Analysis. It's used to draw samples from a distribution that is either hard to sample from or its probability density function (pdf) is only known up to a normalizing constant. The Gibbs sampler algorithm generates a Markov chain which has as its stationary distribution the posterior distribution by simulating observations from a different proposed distribution. This simulation procedure enables us to draw a sample from the posterior distribution that can be used in estimation and other statistical inference. This thesis is organized as follows.

Research paper thumbnail of MannKendallTest

Research paper thumbnail of L-moments of Cubic Transmuted Gompertz-Makeham Distribution

Research paper thumbnail of Simulating confidence intervals for the mean of a normal population

Research paper thumbnail of Moments Equalities for Nonnegative Integer-Valued Random Variables

We present and prove two theorems about equalities for the nth moment of nonnegative integer-valu... more We present and prove two theorems about equalities for the nth moment of nonnegative integer-valued random variables. These equalities generalize the well known equality for the rst moment of a nonnegative integer-valued random variable X in terms of its cumulative distribution function, or in terms of its tail distribution.

Research paper thumbnail of A Mathematical Model for Cluster Applications

IUG Journal of Natural Studies, 2015

In this paper, we use probabilistic modeling and techniques to derive formulas in order to determ... more In this paper, we use probabilistic modeling and techniques to derive formulas in order to determine the number of computer nodes needed to execute applications on a cluster of computers so that application response time can be satisfied. Our probabilistic model is basically an M/G/1/K queueing system. In this model, we account for the workload conditions (in terms of the number of applications or jobs being received per unit time) as well as the processing power of each node. Finally, We use simulations to present a numerical example showing how our probabilistic model can be used.

Research paper thumbnail of Distributions of Gamma m-Spacings

IUG Journal of Natural Studies, 2016

This paper is concerned with the distributions of the -spacings of order statistics associated wi... more This paper is concerned with the distributions of the -spacings of order statistics associated with a sample from a two-parameter gamma population when the shape parameter is a positive integer. We prove that the -spacings have finite mixture distributions with negative components, each of which in turn has a finite gamma mixture distribution. We write the probability density function of the -spacings in closed form and provide a Mathematica code for the implementation.

Research paper thumbnail of On The Spacings of Exponential and Uniform order Statistics

We study the distributions of spacings of order statistics both when the distribution of the rand... more We study the distributions of spacings of order statistics both when the distribution of the random sample has support of the form [a ,b ] and of the form(- ¥ , + ¥ ) . Then we turn to the study of the distributions and some properties of the spacings of the exponential and uniform order statistics. We show that the spacings of adjacent exponential order statistics are independent and exponentially distributed with different parameters. We also prove that these spacings are transformations of some beta distribution in the case of sampling from a uniform population. We also prove some relations between these spacings in the exponential and uniform cases. Order statistics, spacings of order statistics, characterization of distributions, adjacent order statistics. الملخص : ندرس في هذه الورقة توزيع الفراغات بين الإحصائيات المرتبة في كل من الحالتين وعندما يأخذ قيمه [a ,b ] عندما يكون توزيع العينة العشوائية يأخذ قيمه باحتمال موجب في الفترة باحتمال موجب في الفترة ( ¥ + , ¥ -). وبعد ذلك نتح...

Research paper thumbnail of Moments of order statistics from the Gompertz-Makeham Distribution

IUG Journal of Natural Studies, 2019

In this paper we study the moments of order statistics corresponding to a random sample of finite... more In this paper we study the moments of order statistics corresponding to a random sample of finite size from the three-parameter Gompertz-Makeham distribution. We will derive explicit closed-form formulas for the moment-generating function of order statistics from this distribution in terms of the integro-exponential function. Then, we derive explicit close-form formulas for the moments of all orders random variables follow this distribution. Then we prove an identity that relates the moments of order statistics from the Gompertz-Makeham distribution to the moments of random variables from this distribution.

Research paper thumbnail of Higher Rank Transmuted Families of Distributions

IUG Journal of Natural Studies, 2019

In this paper, a generalized method for generating new families of distributions is proposed. The... more In this paper, a generalized method for generating new families of distributions is proposed. The new proposed method generalizes some known methods by introducing higher rank transmutation maps to generate more flexible and tractable lifetime families of probability models. A number of the known methods turn out to be special cases of the proposed method in this paper. Some examples are given to demonstrate how the method works on the Gompertz, exponential, and Weibull probability distributions.

Research paper thumbnail of Bayesian Inference on the Generalized Gamma Distribution using Conjugate Priors

June.2020

This paper focuses on the three-parameter generalized gamma distribution and uses Bayesian techni... more This paper focuses on the three-parameter generalized gamma distribution and uses Bayesian techniques to estimate its parameters. Many authors con-sidered estimating the parameters of the generalized gamma distribution in a Bayesian framework using Jeffrey’s priors. Others used different loss functions and the least squares approach. This study uses Bayesian techniques to estimate the three-parameter generalized gamma distribution by using conjugate priors. The random Metropolis algorithm is used to simulate the Bayesian estimates of the three parameters. Then these estimates are compared to the maximum like-lihood estimates using the mean error through simulation. It has been shown in this paper that the obtained estimates using this approach is more accurate than the traditional methods of estimation such as the Maximum likelihood method. The same approach is then used to estimate the parameters of mixtures of the generalized gamma parameters using conjugate priors.

Research paper thumbnail of A Generalized Transmuted Fre ́chet Distribution

Journal of Statistics Applications & Probability

Research paper thumbnail of Gamma Spacings as Finite Gamma Mixtures

IUG Journal of Natural Studies, 2017

This paper considers the distributions of spacings between successive order statistics correspond... more This paper considers the distributions of spacings between successive order statistics corresponding to a random sample from a two-parameter gamma distribution . We prove that when the shape parameter of the underlying distribution is a positive integer, these spacings can be expressed as finite gamma mixtures. We present exact formulas for computing the distributions of the spacings. Then we present a Mathematica program to implement the results.

Research paper thumbnail of A New Method of Generating Life-Time Distributions

Abstract. Statisticians are interested in improving the ability of classical distributions to app... more Abstract. Statisticians are interested in improving the ability of classical distributions to appropriately fit real-world data and accurately describe its characteristics. Typically, this can be achieved by extending known distributions by incorporating extra parameters or compounding distributions. In this paper, a flexible general method to obtain new families of distributions is proposed. A number of known methods turn out to be special cases of the proposed method in this paper. Some examples are given to demonstrate the power of the proposed method such as the exponential and the Weibull distribution.

Research paper thumbnail of Simulating random walk using Wolfram Mathematica

Research paper thumbnail of Implementing the Moments of Gamma Spacings Using Wolfram Mathematica

Research paper thumbnail of The Cubic Rank Transmuted Gompertz- Makeham Distribution: Properties and L-moments

Research paper thumbnail of Mathematica code for constructing a 100(1-alpha)% confidence interval for the mean of the normal distribution when its variance is known

Research paper thumbnail of Short-Time Asymptotics of the Neumann Heat Kernel for Antipodal Points on the Exterior of a Ball

Research paper thumbnail of Simulating the minimum spacing of exponential order statistics

Research paper thumbnail of Bayesian Inference on Finite Mixtures of Poisson Distributions

Mixed Poisson distributions are widely used in various disciplines to model data in which each ob... more Mixed Poisson distributions are widely used in various disciplines to model data in which each observation is assumed to come from one of a number of Poisson distributions with different parameters. In this thesis, we investigate the Bayesian estimation for the finite Poisson mixture model using the Gibbs sampler as an important one of the MCMC methods. Our approach in this thesis depends on using the Gibbs sampler to simulate a Markov chain which has the posterior density as its stationary distribution. Then we use the resulting sample to make the suitable Bayesian computations and draw conclusion about the unknown parameters of the Poisson mixture model. We conclude this thesis by presenting a real data example to illustrate our methodology. vi Contents Abstract vi pdf probability density function. pmf probability mass function. xii algorithm is one of the most basic Markov Chain Monte Carlo Methods that is used in Bayesian Analysis. It's used to draw samples from a distribution that is either hard to sample from or its probability density function (pdf) is only known up to a normalizing constant. The Gibbs sampler algorithm generates a Markov chain which has as its stationary distribution the posterior distribution by simulating observations from a different proposed distribution. This simulation procedure enables us to draw a sample from the posterior distribution that can be used in estimation and other statistical inference. This thesis is organized as follows.

Research paper thumbnail of MannKendallTest

Research paper thumbnail of L-moments of Cubic Transmuted Gompertz-Makeham Distribution

Research paper thumbnail of Simulating confidence intervals for the mean of a normal population

Research paper thumbnail of Moments Equalities for Nonnegative Integer-Valued Random Variables

We present and prove two theorems about equalities for the nth moment of nonnegative integer-valu... more We present and prove two theorems about equalities for the nth moment of nonnegative integer-valued random variables. These equalities generalize the well known equality for the rst moment of a nonnegative integer-valued random variable X in terms of its cumulative distribution function, or in terms of its tail distribution.

Research paper thumbnail of A Mathematical Model for Cluster Applications

IUG Journal of Natural Studies, 2015

In this paper, we use probabilistic modeling and techniques to derive formulas in order to determ... more In this paper, we use probabilistic modeling and techniques to derive formulas in order to determine the number of computer nodes needed to execute applications on a cluster of computers so that application response time can be satisfied. Our probabilistic model is basically an M/G/1/K queueing system. In this model, we account for the workload conditions (in terms of the number of applications or jobs being received per unit time) as well as the processing power of each node. Finally, We use simulations to present a numerical example showing how our probabilistic model can be used.

Research paper thumbnail of Distributions of Gamma m-Spacings

IUG Journal of Natural Studies, 2016

This paper is concerned with the distributions of the -spacings of order statistics associated wi... more This paper is concerned with the distributions of the -spacings of order statistics associated with a sample from a two-parameter gamma population when the shape parameter is a positive integer. We prove that the -spacings have finite mixture distributions with negative components, each of which in turn has a finite gamma mixture distribution. We write the probability density function of the -spacings in closed form and provide a Mathematica code for the implementation.

Research paper thumbnail of On The Spacings of Exponential and Uniform order Statistics

We study the distributions of spacings of order statistics both when the distribution of the rand... more We study the distributions of spacings of order statistics both when the distribution of the random sample has support of the form [a ,b ] and of the form(- ¥ , + ¥ ) . Then we turn to the study of the distributions and some properties of the spacings of the exponential and uniform order statistics. We show that the spacings of adjacent exponential order statistics are independent and exponentially distributed with different parameters. We also prove that these spacings are transformations of some beta distribution in the case of sampling from a uniform population. We also prove some relations between these spacings in the exponential and uniform cases. Order statistics, spacings of order statistics, characterization of distributions, adjacent order statistics. الملخص : ندرس في هذه الورقة توزيع الفراغات بين الإحصائيات المرتبة في كل من الحالتين وعندما يأخذ قيمه [a ,b ] عندما يكون توزيع العينة العشوائية يأخذ قيمه باحتمال موجب في الفترة باحتمال موجب في الفترة ( ¥ + , ¥ -). وبعد ذلك نتح...

Research paper thumbnail of Moments of order statistics from the Gompertz-Makeham Distribution

IUG Journal of Natural Studies, 2019

In this paper we study the moments of order statistics corresponding to a random sample of finite... more In this paper we study the moments of order statistics corresponding to a random sample of finite size from the three-parameter Gompertz-Makeham distribution. We will derive explicit closed-form formulas for the moment-generating function of order statistics from this distribution in terms of the integro-exponential function. Then, we derive explicit close-form formulas for the moments of all orders random variables follow this distribution. Then we prove an identity that relates the moments of order statistics from the Gompertz-Makeham distribution to the moments of random variables from this distribution.

Research paper thumbnail of Higher Rank Transmuted Families of Distributions

IUG Journal of Natural Studies, 2019

In this paper, a generalized method for generating new families of distributions is proposed. The... more In this paper, a generalized method for generating new families of distributions is proposed. The new proposed method generalizes some known methods by introducing higher rank transmutation maps to generate more flexible and tractable lifetime families of probability models. A number of the known methods turn out to be special cases of the proposed method in this paper. Some examples are given to demonstrate how the method works on the Gompertz, exponential, and Weibull probability distributions.

Research paper thumbnail of Bayesian Inference on the Generalized Gamma Distribution using Conjugate Priors

June.2020

This paper focuses on the three-parameter generalized gamma distribution and uses Bayesian techni... more This paper focuses on the three-parameter generalized gamma distribution and uses Bayesian techniques to estimate its parameters. Many authors con-sidered estimating the parameters of the generalized gamma distribution in a Bayesian framework using Jeffrey’s priors. Others used different loss functions and the least squares approach. This study uses Bayesian techniques to estimate the three-parameter generalized gamma distribution by using conjugate priors. The random Metropolis algorithm is used to simulate the Bayesian estimates of the three parameters. Then these estimates are compared to the maximum like-lihood estimates using the mean error through simulation. It has been shown in this paper that the obtained estimates using this approach is more accurate than the traditional methods of estimation such as the Maximum likelihood method. The same approach is then used to estimate the parameters of mixtures of the generalized gamma parameters using conjugate priors.

Research paper thumbnail of A Generalized Transmuted Fre ́chet Distribution

Journal of Statistics Applications & Probability

Research paper thumbnail of Gamma Spacings as Finite Gamma Mixtures

IUG Journal of Natural Studies, 2017

This paper considers the distributions of spacings between successive order statistics correspond... more This paper considers the distributions of spacings between successive order statistics corresponding to a random sample from a two-parameter gamma distribution . We prove that when the shape parameter of the underlying distribution is a positive integer, these spacings can be expressed as finite gamma mixtures. We present exact formulas for computing the distributions of the spacings. Then we present a Mathematica program to implement the results.