scholarly journals Mathematical Models of Geometric Sizes of Cereal Crops’ Seeds as Dependent Random Variables

2018 ◽  
Vol 21 (3) ◽  
pp. 100-104 ◽  
Author(s):  
Roman Kuzm Inskyi ◽  
Stefan Kovalishyn ◽  
Yurij Kovalchyk ◽  
Roman Sheremeta
Keyword(s):  
Mathematical Models ◽  
Normal Distribution ◽  
Random Variables ◽  
Correlation Coefficients ◽  
Independent Random Variables ◽  
Cereal Crops ◽  
Density Functions ◽  
Dependent Random Variables ◽  
Data Approximation ◽  
Wheat Seeds

Abstract Dimensions of 100 randomly selected wheat seeds of the Smuglyanka variety, rye seeds of the Puhovchanka variety and barley seeds of the Pejas variety were determined by measuring their length (l), width (b) and thickness (h). Results of the measurements were processed by the methods of mathematical statistics; parameters of distributions of individual sizes as random variables were calculated. On the basis of values of variation coefficient, the density function of normal distribution (Gaussian distribution) was taken as a model of individual sizes of seeds. Models of two-dimensional distributions of seed sizes as independent random variables were presented. Correlation coefficients between geometric sizes of seeds were calculated. Obtained values of the correlation coefficients indicate that the geometric sizes of seeds should be considered as dependent random variables. Mathematical models of geometric sizes of studied cereal crops’ seeds as dependent random variables in the form of density functions of their normal distribution were proposed. By values of the sums of squared deviations as a fitting criterion, it was established that the mathematical models of geometric sizes of seeds as dependent random variables in the form of density functions of their normal distribution provide better data approximation than the mathematical models of geometric sizes of some cereal crops’ seeds as independent random variables.

Classes of probability density functions having Laplace transforms with negative zeros and poles

1987 ◽  
Vol 19 (3) ◽  
pp. 632-651 ◽  
Author(s):  
Ushio Sumita ◽  
Yasushi Masuda
Keyword(s):  
Probability Density ◽  
Practical Importance ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Laplace Transforms ◽  
Probability Density Functions ◽  
Independent Random Variables ◽  
Density Functions ◽  
Exponential Random Variables ◽  
Zeros And Poles

We consider a class of functions on [0,∞), denoted by Ω, having Laplace transforms with only negative zeros and poles. Of special interest is the class Ω+ of probability density functions in Ω. Simple and useful conditions are given for necessity and sufficiency of f ∊ Ω to be in Ω+. The class Ω+ contains many classes of great importance such as mixtures of n independent exponential random variables (CMn), sums of n independent exponential random variables (PF∗n), sums of two independent random variables, one in CMr and the other in PF∗1 (CMPFn with n = r + l) and sums of independent random variables in CMn(SCM). Characterization theorems for these classes are given in terms of zeros and poles of Laplace transforms. The prevalence of these classes in applied probability models of practical importance is demonstrated. In particular, sufficient conditions are given for complete monotonicity and unimodality of modified renewal densities.

scholarly journals On Error Bounds for Approximations to Multivariate Distributions II

2002 ◽  
Vol 32 (1) ◽  
pp. 57-69
Author(s):  
Bjørn Sundt ◽  
Raluca Vernic
Keyword(s):  
Error Bounds ◽  
Random Variables ◽  
Independent Random Variables ◽  
Multivariate Distributions ◽  
Dependent Random Variables ◽  
Linear Transforms ◽  
Original Distribution ◽  
Multivariate Bernoulli ◽  
Bernoulli Distributions

AbstractIn the present paper, we study error bounds for approximations to multivariate distributions. In particular, we discuss some general versions of compound multivariate distributions and look at distributions of dependent random variables constructed by linear transforms of independent random variables or vectors. Special attention is paid to the case when the support of the original distribution is restricted. We also look at some applications with multivariate Bernoulli distributions.

Some Inequalities of Partial Sums for Negatively Dependent Random Variables

2012 ◽  
Vol 195-196 ◽  
pp. 694-700
Author(s):  
Hai Wu Huang ◽  
Qun Ying Wu ◽  
Guang Ming Deng
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Exponential Inequalities ◽  
Dependent Random Variables ◽  
Probability Inequalities ◽  
Associated Random Variables ◽  
Negatively Dependent Random Variables

The main purpose of this paper is to investigate some properties of partial sums for negatively dependent random variables. By using some special numerical functions, and we get some probability inequalities and exponential inequalities of partial sums, which generalize the corresponding results for independent random variables and associated random variables. At last, exponential inequalities and Bernsteins inequality for negatively dependent random variables are presented.

Characterization of a class of second-order density functions

1967 ◽  
Vol 4 (1) ◽  
pp. 123-129 ◽  
Author(s):  
C. B. Mehr
Keyword(s):  
Exponential Distribution ◽  
Gamma Distribution ◽  
Gaussian Distribution ◽  
Random Variables ◽  
Second Order ◽  
Independent Random Variables ◽  
Density Functions ◽  
Linear Filtering ◽  
Non Linear

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.

scholarly journals Bernstein-Type Inequality for Widely Dependent Sequence and Its Application to Nonparametric Regression Models

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Aiting Shen
Keyword(s):  
Strong Consistency ◽  
Nearest Neighbor ◽  
Type Inequality ◽  
Random Variables ◽  
Independent Random Variables ◽  
Bernstein Type ◽  
Dependent Random Variables ◽  
Bernstein Type Inequality ◽  
Dependent Sequence ◽  
Fixed Design Regression

We present the Bernstein-type inequality for widely dependent random variables. By using the Bernstein-type inequality and the truncated method, we further study the strong consistency of estimator of fixed design regression model under widely dependent random variables, which generalizes the corresponding one of independent random variables. As an application, the strong consistency for the nearest neighbor estimator is obtained.

On the asymptotic independent representations for sums of some weakly dependent random variables

2006 ◽  
Vol 43 (1) ◽  
pp. 33-46
Author(s):  
Rafik Aguech ◽  
Sana Louhichi ◽  
Sofyen Louhichi
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Triangular Array ◽  
Independent Random Variables ◽  
Dependent Random Variables ◽  
Limiting Behavior ◽  
Weakly Dependent Random Variables ◽  
Weakly Dependent

Let, for each n?N, (Xi,n)0?i?nbe a triangular array of stationary, centered, square integrable and associated real valued random variables satisfying the weakly dependence condition lim N?N0limsup n?+8nSr=NnCov (X0,n, Xr,n)=0;where N0is either infinite or the first positive integer Nfor which the limit of the sum nSr=NnCov (X0,n, Xr,n) vanishes as n goes to infinity. The purpose of this paper is to build, from (Xi,n)0?i?n, a sequence of independent random variables (X˜i,n)0?i?nsuch that the two sumsSi=1nXi,nandSi=1nX˜i,nhave the same asymptotic limiting behavior (in distribution).

Limit distributions of the Pearson statistics for nonhomogeneous polynomial scheme

2019 ◽  
Vol 29 (4) ◽  
pp. 233-239
Author(s):  
Maxim P. Savelov
Keyword(s):  
Quadratic Form ◽  
Normal Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Standard Normal Distribution ◽  
Limit Distributions ◽  
Standard Normal ◽  
Polynomial Scheme

Abstract For a nonhomogeneous polynomial scheme, conditions are found under which the Pearson statistic distributions converge to the distribution of nonnegative quadratic form of independent random variables with the standard normal distribution.

scholarly journals Moment inequality of the minimum for nonnegative negatively orthant dependent random variables

Filomat ◽  
2014 ◽  
Vol 28 (7) ◽  
pp. 1475-1481
Author(s):  
Xuejun Wang ◽  
Shijie Wang ◽  
Shuhe Hu
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Exponential Inequality ◽  
Moment Inequalities ◽  
Moment Inequality ◽  
Dependent Random Variables ◽  
The Moment

Let {xn,n ? 1} be a sequence of positive numbers and {?n,n ? 1} be a sequence of nonnegative negatively orthant dependent (NOD) random variables satisfying certain distribution conditions. An exponential inequality for the minimum min1?i?n xi?i is given. In addition, the moment inequalities of the minimum (Ek - min1?i?n|xi?i|p)1/p for nonnegative negatively orthant dependent random variables are established, where p > 0 and k = 1,2,..., n. Our results generalize the corresponding ones for independent random variables to the case of negatively orthant dependent random variables.

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