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Statistical modeling is fundamentally based on probability distributions, which can be discrete or continuous and univariate or multivariate. This review focuses on the methods used to construct these distributions, covering both traditional and newly developed approaches. We first examine classic distributions such as the normal, exponential, gamma, and beta for univariate data, and the multivariate normal, elliptical, and Dirichlet for multidimensional data. We then address how, in recent decades, the demand for more flexible modeling tools has led to the creation of complex meta-distributions built using copula theory.

In this paper a methodology is developed to solve a nonlinear interval optimization problem by transforming this to a general optimization problem which is free from interval uncertainty. To address the interval uncertainty, relation between an interval and a random variable is established according to the 3 sigma-rule. Using this relation an interval function is associated with a function of random variables and an interval inequality is associated with a chance constraint. The interval optimization problem is then transformed into a nonlinear stochastic programming problem. Further, the existence of a preferable solution of the original problem is established using Chance Constrained Programming technique.

This article provides an alternative method of finding the distribution of a function of one or more random variables using the Dirac generalized function. Unlike the conventional change-of-variable technique which involves a one-to-one transformation and computation of the Jacobian, here the procedure for obtaining the distributions of functions of random variables is shown to be simple, direct, and powerful.

The exact distributions of the quotients X / Y and Y / ( X + Y ) when X and Y are independent and triangularly distributed random variables are obtained. These quotients are useful especially in operations research and reliability engineering, and some reliability applications of the results are also given.

The chief attraction of mathematics is its beauty. It is customary, nonetheless, to catalog the usefulness of mathematics. Accordingly, this paper shows how aspects of three quite different sociological problems--assortative mating and the rise of gender inequality, the sense of justice, and economic and political upheavals and societal transformations--have an identical underlying mathematical structure. Thus, progress in understanding the three sets of phenomena is linked; any new result obtained for one problem will shed light on the other two.

Processes of beneficiation are a primary component of mineral processing operations. The efficiency of the process and potential beneficiation of the material are evaluated by means of beneficiation curves. There are many varieties of beneficiation curves, among which one of the most often applied is a group called Henry's beneficiation curves. Of these, the basic curve for ash content in feed is the most often used. This paper presents the methods of ash content curve approximation. This paper is divided into two parts. Part one contains the approximation of the basic Henry's beneficiation curve λ = λ(γ) for energetic coal of type 31 (data from one of the Upper Silesian coal mines were applied for this purpose) conducted by means of two varying methods. The first method was based on a determination of the functional relation between ash content and density λ = λ[0](ρ) as well as between yield and density γ = F(ρ). Then, by determination of the reverse function ρ = F[-1](γ) and its combination with the first function, the searched relation for λ(γ) was obtained. The second method was based on approximation of the function γ = Φ(λ) by means of combining two logistical functions (the combining point was determined by means of the ordinary kriging method). Then, using the reverse function, the searched function λ(γ) was obtained. The adequacy of approximation was evaluated by comparison of the ash content both in individual fractions and throughout the whole material. The second part of the paper proposes ways of determining the surface λ = λ(d,ρ) by means of regression of the first type by applying two- and three-dimensional Morgenstern distribution functions and by means of regression of the second type by applying known two-dimensional distribution functions or two-dimensional kernel approximation. The latter function is the nonparametric statistical method increasingly used. Furthermore, the paper provides formulas for mean ash content as a component of the whole investigated material.

The need to establish the independence of the sample mean and the sample variance in sampling from a normal population arises early in a course in statistics. For the result is an essential ingredient in the derivation of the Student-t distribution for statistical inference. Often this need arises before the tools, notably multivariate methods, for a rigorous proof are available. Occasionally one will find attempts to derive this result using only bivariate assumptions. A recent article in this journal, as well as some current textbooks, offer such a proof. In all cases there are serious questions about the validity of the proofs.

This chapter contains sections titled: Introduction Complex Random Variables The Characteristic Function Characteristic Function of a Transformed Random Variable Characteristic Function of a Multidimensional Random Variable The Generating Function Several Jointly Gaussian Random Variables Spherically Symmetric Vector Random Variables Entropy Associated with Random Variables Copulas Sequences of Random Variables Convergent Sequences and Laws of Large Numbers Convergence of Probability Distributions and the Central Limit Theorem Summary

This chapter contains sections titled: Introduction Some Properties of a Random Variable Higher Moments Expectation of a Function of a Random Variable The Variance of a Function of a Random Variable Bounds on the Induced Distribution Test Sampling Conditional Expectation with Respect to an Event Covariance and Correlation Coefficient The Correlation Coefficient as Parameter in a Joint Distribution More General Kinds of Dependence Between Random Variables The Covariance Matrix Random Variables as the Elements of a Vector Space Estimation The Stieltjes Integral Summary

The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth. In this present paper we consider the normal matrix model with cubic plus linear potential. In order to regularize the model, we follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain We also study in detail the mother body problem associated to To construct the mother body measure, we define a quadratic differential Following previous works of Bleher & Kuijlaars and Kuijlaars & López, we consider multiple orthogonal polynomials associated with the normal matrix model. Applying the Deift-Zhou nonlinear steepest descent method to the associated Riemann-Hilbert problem, we obtain strong asymptotic formulas for these polynomials. Due to the presence of the linear term in the potential, there are no rotational symmetries in the model. This makes the construction of the associated