# T-X Family of Distributions

## Top PDF T-X Family of Distributions: ### On generating T-X family of distributions using quantile functions

The cumulative distribution function (CDF) of the T - X family is given by R { W ( F ( x ))}, where R is the CDF of a random variable T , F is the CDF of X and W is an increasing function defined on [0, 1] having the support of T as its range. This family provides a new method of generating univariate distributions. Different choices of the R , F and W functions naturally lead to different families of distributions. This paper proposes the use of quantile functions to define the W function. Some general properties of this T - X system of distributions are studied. It is shown that several existing methods of generating univariate continuous distributions can be derived using this T - X system. Three new distributions of the T-X family are derived, namely, the normal-Weibull based on the quantile of Cauchy distribution, normal-Weibull based on the quantile of logistic distribution, and Weibull-uniform based on the quantile of log-logistic distribution. Two real data sets are applied to illustrate the flexibility of the distributions. ### Exponentiated Marshall-Olkin family of distributions

In the past few years, several ways of generating new distributions from classic ones were developed and discussed. Eugene et al. (2002) defined a class of beta-generated dis- tribution. Jones (2004) studied a family of distributions that arises naturally from the distribution of the order statistics and introduced general properties of the proposed class of distributions. Zografos and Balakrishnan (2009) proposed the gamma-generated family of distributions. Later, Cordeiro and de Castro (2011) defined the Kumaraswamy family. Recently, Alzaatreh et al. (2013) proposed a new technique to derive wider families by using any probability density function (pdf ) as a generator. This generator called the T-X family of distributions has cumulative distribution function (cdf ) defined by ### A General Transmuted Family of Distributions

The quadratic transmuted family of distributions by Shaw and Buckley (2007), has opened doors to extend the existing probability models to capture the quadratic behavior of the data. The quadratic transmuted class has been further studied and extended by Aryal and Tsokos (2009, 2011); Nofal et al. (2017); Alizadeh et al. (2016); Merovci et al. (2016) and Yousof et al. (2015). At this time, quadratic transmuted distributions are familiar in the literature, several quadratic transmuted distributions are provided by Tahir and Cordeiro (2016). Alizadeh et al. (2017), have noted that the quadratic transmuted family of distribution can be linked with the T-X family of distributions for suitable choice of a bounded density function between [0, 1]. In order to capture the complexity of the data and increases the flexibility, new classes of cubic transmuted distributions have been developed by Granzotto et al. (2017); AL-Kadim and Mohammed (2017). ### The Nadarajah Haghighi Topp Leone-G Family of Distributions ‎with Mathematical Properties and Applications

= +   (2) Based on the idea of T-X family pioneerd by Alzaatreh et al. (2013), TL-G class and the Nadarajah Haghighi distribution, we introduce a new class of continuous distributions called the Nadarajah Haghighi Topp Leone-G (NHTL-G for short) family with cdf given by ### New General Transmuted Family of Distributions with Applications

Theorem 3.2. Let G(x) be cdf of a random variable X and p(t) be pdf of a bounded random variable T with support on [0,1], then cubic transmuted family given in (7) can be obtained by using TX family of distributions, introduced by Alzaatreh et al. (2013), for suitable choice of p(t). Also, the pdf p(t) can be written as weighted sum of three bounded densities p 1 (t), p 2 (t) and p 3 (t) with support on [0,1]. ### A new Weibull-X family of distributions: properties, characterizations and applications

In the field of reliability theory, modeling of lifetime data is very crucial. A number of statistical distributions such as Weibull, Pareto, Gompertz, linear failure rate, Rayleigh, Exonential etc., are available for modeling lifetime data. However, in many practical areas, these classical distributions do not provide adequate fit in modeling data, and there is a clear need for the extended version of these classical distributions. In this re- gard, serious attempts have been made to propose new families of continuous prob- ability distributions that extend the existing well-known distributions by adding additional parameter(s) to the model of the baseline random variable. The well-known family of distributions are: the beta-G by Eugene et al. (2002), Jones (2004), Gamma-G (type-1) due to Zografos and Balakrishnan (2009), Mc-G proposed by Alexander et al. (2012), Log-Gamma-G Type-2 of Amini et al. (2012), Gamma-G (type-2) studied by Risti’c and Balakrishnan (2012), Gamma-G (type-3) of Torabi and Montazeri (2012), Weibull-X family of distributions of Alzaatreh et al. (2013), exponentiated generalized class of Cordeiro et al. (2013), Logistic-G introduced by Torabi and Montazeri (2014), Gamma-X family of Alzaatreh et al. (2014), odd generalized exponential-G of Tahir et al. (2015a, b), type I half-logistic family of Cordeiro et al. (2016), Kumaraswamy Weibull-generated family of Hassan and Elgarhy (2016), new Weibull-G family of Tahir et al. (2016), generalized transmuted-G of Nofal et al. (2017) and a new generalized family of distributions of Ahmad (2018). Let v(t) be the probability density function ### Truncated log-logistic Family of Distributions

exponentiated generalized by Cordeiro et al. (6), transformed-transformer (T-X) by Alzaatreh et al. (7), exponentiated T-X by Alzaghal et al. (8), odd Weibull-G by Bourguignon et al. (9), exponentiated half- logistic by Cordeiro et al. (10), T-X{Y}- quantile based approach by Aljarrah et al. (11), Lomax-G by Cordeiro et al. (12), Kumaraswamy-G class of distributions by Cordeiro et al. (13), Kumaraswamy odd log- logistic-G by Alizadeh et al. (14), logistic-X by Tahir et al. (15) and alpha power transformation family of distributions introduced by Mahdavi and Kundu (16). ### Generalized Topp-Leone family of distributions

transformed-transformer (T-X) by Alzaatreh et al. (7), exponentiated T-X by Alzaghal et al. (8), odd Weibull-G by Bourguignon et al. (9), exponentiated half-logistic by Cordeiro et al. (10), T-X{Y} quantile-based approach by Aljarrah et al. (11), T-R{Y} by Alzaatreh et al. (12), Lomax- G by Cordeiro et al. (13), Kumaraswamy-G class of distributions by Cordeiro et al. (14), Kumaraswamy odd log-logistic-G by Alizadeh et al. (15), logistic-X by Tahir et al. (16) and alpha power transformation family of distributions introduced by Mahdavi and Kundu (17). Recently, Topp-Leone generated (TL-G) family of distributions proposed by Al-Shomrani et al. (18) with its cumulative distribution function (CDF) and probability density function (PDF) as follows: ### T-normal family of distributions: a new approach to generalize the normal distribution

The idea of generating skewed distributions from normal has been of great interest among researchers for decades. This paper proposes four families of generalized normal distributions using the T-X framework. These four families of distributions are named as T-normal families arising from the quantile functions of (i) standard exponential, (ii) standard log-logistic, (iii) standard logistic and (iv) standard extreme value distributions. Some general properties including moments, mean deviations and Shannon entropy of the T-normal family are studied. Four new generalized normal distributions are developed using the T-normal method. Some properties of these four generalized normal distributions are studied in detail. The shapes of the proposed T-normal distributions can be symmetric, skewed to the right, skewed to the left, or bimodal. Two data sets, one skewed unimodal and the other bimodal, are fitted by using the generalized T-normal distributions. ### Odds Generalized Exponential-Pareto Distribution: Properties and Application

In the generalized class of beta distribution, since the beta random variable lies between 0 and 1, and the distribution function also lies between 0 and 1, to find out cdf of generalized distribution, the upper limit is replaced by cdf of the generalized distribution. Alzaatreh et al. (2013) has proposed a new generalized family of distributions, called T-X family, and the cumulative distribution function (cdf) is defined as ### The generalized Cauchy family of distributions with applications

Hereafter, the family of distributions in (5) will be called the T-Cauchy{Y} family. It is clear that the PDF in (5) is a generalization of Cauchy distribution. From (1), if T ¼ d Y ; then X ¼ d Cauchy ð Þ: θ Also, if Y ¼ d Cauchy ð Þ; θ then X ¼ d T : Furthermore, when T ~ beta(a, b) and Y ~ uniform(0, 1), the T-Cauchy{Y} reduces to the beta- Cauchy distribution (Alshawarbeh et al. 2013). When T ~ Power(a) and Y ~ uniform(0, 1), the T-Cauchy{Y} reduces to the exponentiated-Cauchy distribution (Sarabia and Castillo 2005). Table 1 gives six quantile functions of known distributions (in standard form) which will be applied to generate T-Cauchy{Y} sub-families in the following subsections. It is straightforward to use non-standard quantile functions. By using non-standard quan- tile functions, many resulting distributions in the T-R{Y} family will have more than five parameters, which are not practically useful (Johnson et al. 1994, p. 12). Thus, we focus on the standard quantile functions in this paper. ### The Transmuted Exponentiated Generalized-G Family of Distributions

The rest of the paper is outlined as follows. In Section 2, we define the TExG-G family of distributions and provide its special models. In Section 3, we derive a very useful linear representation for the TExG-G density function. Three special models of this family are presented in Section 4 and some plots of their pdf's are given. We obtain in Section 5 some general mathematical properties of the proposed family including asymptotics, extreme values, ordinary and incomplete moments, probability weighted moments (PWMs), mean deviations, residual life function and reversed residual life function. Order statistics and their moments are investigated in Section 6. In Section 7, we determine the stress-strength model for the proposed family. In Section 8, some characterizations results are provided. Maximum likelihood estimation (MLE) of the model parameters is investigated in Section 9. In Section 10, we perform an application to a real dataset to illustrate the potentiality of the new family. Section 11 deals with a small simulation study to assess the performance of the MLE method. Finally, some concluding remarks are presented in Section 12. ### Recent Developments in Distribution Theory: A Brief Survey and Some New Generalized Classes of distributions

The objectives of the present study are three-fold: Firstly, we present an up-to-date account of the extended classes of distributions for the readers of modern distribution theory. Secondly, this survey will motivate the researchers to fill up the gap and to furnish their work in remaining applied areas. Thirdly, we propose some new classes of distributions which might be helpful as a tutorial to the beginners of the generalized modeling art. ### Covariance Operators and the Central Limit Theory for "Loop" Markov Chains

This paper is organized as follows. In Section 2, we define the class of Weibull and generalized Weibull distributions, and demonstrate the many existing models that can be deduced as special cases of the proposed unified model. In Section 3, we define the GEWPS class of distributions in terms of distribution functions and special cases of some existing classes. In Section 4, we provide the general properties of the GEWPS class, including the densities and thesurvival and hazard rate functions. Quantiles, moments, and order statistics of GEWPS are discussed in Section 5. The estimation of the GEWPS parameters is investigated in Section 6 using the maximum likelihood method withexpectation-maximization(EM) algorithm and a large sample inference. In Section 7, special subclasses and some special distributions are introduced along with the flexible mathematical forms of theirproperties. In Section 8, two models are presented and applied to illustrate how to use the proposed family.Finally, some concluding remarks are addressed in Section 9. ### A quasi experimental study to assess the effectiveness of structured teaching programme on knowledge regarding post natal care among primi mothers in kannivadi block phc at Dindigul district

Rajan E. (2014) was conducted a effectiveness of self instructional module on knowledge of post natal care for Primi mothers in selected hospitals, Mangalore. The study was conducted on Effectiveness of self instructional module on knowledge of post natal care for Primi mothers in selected hospitals, Mangalore. The research design was a one group Pre-test Post-test design which was a pre experimental research design. 40 mothers were selected by purposive sampling. The pretest knowledge questionnaire was administered to the mothers two days prior to self instructional module, followed by a self instructional module on post natal care. Post-test was conducted after 5 days using the same tool. The collected data were analyzed using descriptive and inferential statistics. The mean knowledge score was 14.98 whereas maximum possible score was 30. Among the 11 areas, the mean percentage knowledge score in the area of caesarean section and self care was 77.50% bladder and bowel care was 60% breast feeding was 58.40% diet was 52.50% pain management was 47.50% post operative complications and home care was 46% baby care was 44.33% early ambulation and exercise was 44% perineal hygiene was 41% wound care was 40.67% and deep breathing and coughing was 40.67%. The 't' value showed significant in the Post-test ('t' calculated value of pretest and Post-test knowledge scores = 18.000, p<0.001) which showed that self-instructional module was effective in improving the knowledge of mothers on post natal care. There was significant association between the level of knowledge and demographic variables namely age parity, education, occupation, monthly income, exposure to health awareness and history of caesarean section. ### A Family of Loss Distributions with an Application to the Vehicle Insurance Loss Data

This article is organized as follows: In Section 2, we define the W-Loss distri- bution and provide some plots for its pdf. We provide in Section 3 some general mathematical properties of the F-Loss distributions. The maximum likelihood es- timates (MLEs) of the unknown parameters and simulation study are presented in Section 4. In Section 5, certain characterizations of the proposed model are provided. The proposed W-Loss distribution is applied to the vehicle insurance loss data in Section 6. Finally, the article is concluded in Section 7. ### The odd generalized exponential family of distributions with applications

The generalized continuous distributions have been widely studied in the literature. We propose a new class of distributions called the odd generalized exponential family. We study some structural properties of the new family including an expansion for its den- sity function. We obtain explicit expressions for the moments, generating function, mean deviations, quantile function and order statistics. The maximum likelihood method is employed to estimate the family parameters. We fit three special models of the proposed family to two real data sets to demonstrate the usefulness of the family. We use four goodness-of-fit statistics in order to verify which distribution provides better fit to these data. We conclude that these three special models provide consistently better fits than other competing models. We hope that the proposed family and its generated models will attract wider applications in several areas such as reliability engineering, insurance, hydrology, economics and survival analysis. ### The Marshall-Olkin extended Weibull family of distributions

We introduce a new class of models called the Marshall-Olkin extended Weibull family of distributions based on the work by Marshall and Olkin (Biometrika 84:641–652, 1997). The proposed family includes as special cases several models studied in the literature such as the Marshall-Olkin Weibull, Marshall-Olkin Lomax, Marshal-Olkin Fréchet and Marshall-Olkin Burr XII distributions, among others. It defines at least twenty-one special models and thirteen of them are new ones. We study some of its structural properties including moments, generating function, mean deviations and entropy. We obtain the density function of the order statistics and their moments. Special distributions are investigated in some details. We derive two classes of entropy and one class of divergence measures which can be interpreted as new goodness-of-fit quantities. The method of maximum likelihood for estimating the model parameters is discussed for uncensored and multi-censored data. We perform a simulation study using Markov Chain Monte Carlo method in order to establish the accuracy of these estimators. The usefulness of the new family is illustrated by means of two real data sets. ### The T box family

At the cellular level, genetic mutations have provided clues about the requirement for T-box genes in a variety of develop- mental processes, but the exact function of T-box genes, including questions of genetic redundancy, still needs to be established. For example, in ulnar-mammary syndrome patients it is not clear why there is no corresponding defect in the legs, a region that normally expresses Tbx3 [30,31]. This may perhaps be due to a redundant function with other T-box genes expressed in the hindlimb, such as Tbx2 or Tbx4. In order to dissect the function of individual T-box genes, it will also be necessary to generate allelic series (as has been useful for the study of Holt-Oram syndrome) and condi- tional mutations. A case for the latter has already been demonstrated for Eomesodermin, a gene expressed in all vertebrates just prior to gastrulation in the prospective mesoderm and, in the mouse, in the trophectoderm, an extraembryonic tissue that is required for placenta forma- tion and is thus unique to mammals . Mice lacking Eomesodermin fail at, or shortly after, implantation, because of a defect in the trophectoderm. This phenotype can be rescued by wild-type trophectoderm, even if the embryo itself is mutant, but when embryonic tissues lack Eomesodermin, mesoderm differentiation and migration fails completely . ### Small x Asymptotics of the Quark and Gluon Helicity Distributions

differences in transverse coordinates by the abbreviated notation x 10 ≡ x 1 − x 0 . The center-of-mass energy squared for the scattering process is s, the infrared (IR) transverse momentum cutoff is Λ, and z is the fraction of the light-cone momentum of the dipole carried by the polarized (anti-)quark. As is well-known, the TMD (3) contains a process-dependent gauge link U [0, r]. For specificity, in  we considered semi-inclusive deep inelastic scattering (SIDIS), although the resulting small-x evolution equations also apply to the collinear quark helicity distribution, which is process independent.