#### International Journal of Statistics and Probability

Journal Information

ISSN / EISSN :
1927-7032 / 1927-7040

Current Publisher: Canadian Center of Science and Education (10.5539)

Total articles ≅ 556

Archived in

SHERPA/ROMEO

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#### Latest articles in this journal

Published: 6 June 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n4p21

**Abstract:**

The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles using computer-generated data based on Newton’s law of motion. To achieve this, this paper derives the probability density function ψ^ab(u_a;v_a,v_b) of the speed u_a of the particle with mass M_a after the collision of two particles with mass M_a in speed v_a and mass M_b in speed v_b. The function ψ^ab(u_a;v_a,v_b) is obtained through a unique procedure that considers (1) the randomness of the relative direction before a collision by an angle α. (2) the randomness of the direction after the collision by another independent angle β. The function ψ^ab(u_a;v_a,v_b) is used in the equation below for the numerical iterations to get new distributions P_new^a(u_a) from old distributions P_old^a(v_a), and repeat with P_old^a(v_a)=P_new^a(v_a), where n_a is the fraction of particles with mass M_a. P_new^1(u_1)=n_1 ∫_0^∞ ∫_0^∞ ψ^11(u_1;v_1,v’_1) P_old^1(v_1) P_old^1(v’_1) dv_1 dv’_1 +n_2 ∫_0^∞ ∫_0^∞ ψ^12(u_1;v_1,v_2) P_old^1(v_1) P_old^2(v_2) dv_1 dv_2 P_new^2(u_2)=n_1 ∫_0^∞ ∫_0^∞ ψ^21(u_2;v_2,v_1) P_old^2(v_2) P_old^1(v_1) dv_2 dv_1 +n_2 ∫_0^∞ ∫_0^∞ ψ^22(u_2;v_2,v’_2) P_old^2(v_2) P_old^2(v’_2) dv_2 dv’_2 The final distributions converge to the Maxwell-Boltzmann speed distributions. Moreover, the square of the root-mean-square speed from the final distribution is inversely proportional to the particle masses as predicted by Avogadro’s law.

Published: 1 June 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n4p33

**Abstract:**

A new distribution called the Lindley-Burr XII (LBXII) distribution is proposed and studied. Some structural properties of the new distribution including moments, conditional moments, distribution of the order statistics and R´enyi entropy are derived. Maximum likelihood estimation technique is used to estimate the model parameters. A simulation study to examine the bias and mean square error of the maximum likelihood estimators is presented and applications to real data sets in order to illustrate the usefulness of the new distribution are given.

Published: 1 June 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n4p52

**Abstract:**

The aim of the study is to analyze the pattern of Gross domestic product (GDP) according to Human development index (HDI) for 184 countries of the world. GDP per capita indicates only economic prosperity but not the overall development of the citizens of a country. This research tries to find out the beneath relationship of the financial state and human development of countries using the data of 2018. For demonstrating this analysis several parametric and non-parametric regression methods subject to shape restriction have been used. The study targets to shed light on comparative performance of shape constrained regression with cone projection, polynomial regression, LOESS, Istonic regression with pooled adjacent violators algorithm, Kernel regression, smoothing spline and generalized additive model in convex situation.

Published: 27 May 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n4p10

**Abstract:**

We are interested in describing the dynamics of the infected size of the SIS Epidemic model using the Birth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) model is defined within a population of constant size $M$; the size is kept constant by replacing each death with a newborn healthy individual. The life span of each individual in the population is modelled by an exponential distribution with parameter $\alpha$; the disease spreads within the population is modelled by a Poisson process with a rate $\lambda_{I}$. $\lambda_{I}=\beta I(1-\frac{I}{M}) $ is similar to the instantaneous rate in the logistic population growth model. The analysis is focused on the disease outbreak, where the reproduction number $R=\frac{\beta} {\alpha} $ is greater than one. As methodology, we use both numerical and analytical approaches. The numerical approach shows that the infected size dynamics converge to a stationary stochastic process. And the analytical results determine the distribution of the stationary stochastic process as a normal distribution with mean $(1-\frac{1}{R}) M$ and Variance $\frac{M}{R} $ when $M$ becomes larger.

Published: 16 May 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n4p1

**Abstract:**

Here in this paper, we investigate the performance of a diagnostic test based on a mixture Gaussian Copula which incorporates a Markov Chain. Suppose that in the context of an infectious disease, there are three states; Susceptible (S), Infected (I), or Recovered (R). We compare the performance of this approach with the ROC (Receiver Operating Characteristic) Curve which is usually used in diagnostic studies.

Published: 29 April 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n3p164

**Abstract:**

Reviewer Acknowledgements for International Journal of Statistics and Probability, Vol. 10, No. 3, 2021

Published: 29 April 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n3p154

**Abstract:**

Propensity Score Methodology (PSM) was used to investigate the effect of education on attitude towards domestic violence (ATDV) among men and women in Nigeria. A total of 14,495 and 33,419 records were extracted for men and women respectively from the 2016-2017 Multiple Indicator Cluster Survey (MICS) in Nigeria. The outcome variable was ATDV. The study framework described the role of education on ATDV in the light of demographic characteristics, socioeconomic profile, and lifestyle. Selection bias was checked among the levels of education using the multinomial logit regression. Propensity scores (PS) and PS weights were generated for the treatment variable and average treatment effects (ATE) of ATDV were estimated using logistic regression that combined regression adjustment and inverse-probability weight. Descriptive statistics, odds ratios and 95%CI were presented. The mean age of men and women were 30.8±10.2 years and 29±9.4 years respectively. About 22% men and 35% women justified domestic violence (DV) respectively. Selection bias was found between the covariates and level of education (p<0.05). PSM effectively corrected the selection bias (SD diff ≈ 0, Variance ratio ≈ 1). Men (AOR = 0.84, 95% CI: 0.78, 0.92) and women (AOR=0.94, 95%CI: 0.80, 2.22) who have attained tertiary level of education were less likely to justify DV in comparison to their uneducated counterparts. Tertiary education was protective for ATDV among men and women. The use of PSM effectively controlled for selection bias in estimating the effect of education on ATDV. PSM will enable researchers make causal inference from non-experimental/cross-sectional studies in situations where randomized control trials are not feasible.

Published: 27 April 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n3p135

**Abstract:**

The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. The Maxwell-Boltzmann speed distribution of mixed particles is based on kinetic theory; however, it has never been derived from a mechanical point of view. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles based on probability analysis and Newton’s law of motion. This paper requires the probability density function (PDF) ψ^ab(u_a; v_a, v_b) of the speed u_a of the particle with mass M_a after the collision of two particles with mass M_a in speed v_a and mass M_b in speed v_b . The PDF ψ^ab(u_a; v_a, v_b) in integral form has been obtained before. This paper further performs the exact integration from the integral form to obtain the PDF ψ^ab(u_a; v_a, v_b) in an evaluated form, which is used in the following equation to get new distribution P_new^a(u_a) from old distributions P_old^a(v_a) and P_old^b(v_b). When P_old^a(v_a) and P_old^b(v_b) are the Maxwell-Boltzmann speed distributions, the integration P_new^a(u_a) obtained analytically is exactly the Maxwell-Boltzmann speed distribution. P_new^a(u_a)=∫_0^∞ ∫_0^∞ ψ^ab(u_a;v_a,v_b) P_old^a(v_a) P_old^b(v_b) dv_a dv_b, a,b = 1 or 2 The mechanical proof of the Maxwell-Boltzmann speed distribution presented in this paper reveals the unsolved mechanical mystery of the Maxwell-Boltzmann speed distribution since it was proposed by Maxwell in 1860. Also, since the validation is carried out in an analytical approach, it proves that there is no theoretical limitation of mass ratio to the Maxwell-Boltzmann speed distribution. This provides a foundation and methodology for analyzing the interaction between particles with an extreme mass ratio, such as gases and neutrinos.

Published: 24 April 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n3p126

**Abstract:**

The copula function is an effective and elegant tool useful for modeling dependence between random variables. Among the many families of this function, one of the most prominent family of copula is the Archimedean family, which has its unique structure and features. Most of the copula functions in this family have only a single dependence parameter which limits the scope of the dependence structure. In this paper we modify the generator of Archimedean copulas in a way which maintains membership in the family while increasing the number of dependence parameters and, consequently, creating new copulas having more flexible dependence structure.

Published: 20 April 2021

International Journal of Statistics and Probability, Volume 10; doi:10.5539/ijsp.v10n3p93

**Abstract:**

In this paper, we present a review on the log-logistic distribution and some of its recent generalizations. We cite more than twenty distributions obtained by different generating families of univariate continuous distributions or compounding methods on the log-logistic distribution. We reviewed some log-logistic mathematical properties, including the eight different functions used to define lifetime distributions. These results were used to obtain the properties of some log-logistic generalizations from linear representations. A real-life data application is presented to compare some of the surveyed distributions.