scholarly journals Possibility of using CMS Maple to study laws of distribution of random variables

Radiotekhnika ◽  
2021 ◽  
pp. 128-134
Author(s):  
I. Moshchenko ◽  
O. Nikitenko ◽  
Yu.V. Kozlov
Keyword(s):  
Data Processing ◽  
Random Variables ◽  
Random Variable ◽  
Large Set ◽  
Practical Applications ◽  
Symbolic Calculations ◽  
Variable Distribution ◽  
Topic Distribution ◽  
Distribution Laws ◽  
Independent Work

The use of CMS Maple for students' practical and independent work is described. The study of random variable distribution laws is actual. Statistical calculations without computer are difficult and require many functional and quintiles tables of standard distributions. This does not contribute to feeling the element of novelty in the material being studied, to be able to arbitrarily change the conditions of tasks, etc., it takes a lot of time in solving applied production problems, which is inappropriate Thus to determine and research random variable distribution laws both in practical applications and in studying we must use special mathematical packages. The most extended of them are Mathcad, MatLab, Mathematica, Maple. Specialized statistical packages (SAS, SPSS, STATISTIKA, STATGRAPHICS) are not relevant to study. Their use for studying requires very high education level in mathematical statistics. Most of the existing math packages allow users to operate at random variables, including the Computer Mathematics System (CMS) Maple. Thus, the purpose of this article is a description of the studying possibilities of the random variables distribution laws with CMS Maple and the application of the acquired skills to the independent work of students. The Maple Statistics Library has a large set of commands for analyzing data, computing various numerical characteristics of random variables, graphing their distribution laws, and for statistical data processing. Thanks to a powerful set of statistical tools, the possibility of symbolic calculations and data processing of CMS Maple, wide possibilities of graphical interpretation of the results obtained not only in a static but also in a dynamic form, it is advisable to use it when studying the topic "Distribution Laws of Random Variables" in students' practical and independent work to use their acquired skills in solving applied problems of science and technology.

scholarly journals ANALYTICAL EVALUATION OF PROCESSING TIME OF A QUERY IN AN INFORMATION SYSTEM

2018 ◽  
pp. 69-73 ◽  
Author(s):  
M. M. Butaev ◽  
A. A. Tarasov
Keyword(s):  
Information Systems ◽  
Normal Distribution ◽  
Processing Time ◽  
Random Variables ◽  
Initial Approximation ◽  
Random Variable ◽  
Sequential Processing ◽  
Processing Times ◽  
Distribution Laws ◽  
Interval Distribution

The normal distribution of a random variable is usually used in studies of the probabilistic characteristics of information systems. However, the approximation by the normal distribution of distributions determined on a limited interval distorts the physical meaning of the model and the numerical results, and it can only be used as an initial approximation. The aim of the work is to improve the methods for calculating the probabilistic characteristics of information systems. The object of the study is an analytical method for calculating the processing time of the query in the system. The subject of the study are formulas for calculating the duration of sequential processing of the query by elements of the system with uniformly distributed random processing times. In deriving the formulas for calculating the probability characteristics of a sum of independent uniformly distributed random variables, the methods of the theory of probability and statistics are applied. It is proposed for random variables, determined only on the positive coordinate axis, to use finite-interval distribution laws, for example, beta distribution. Density formulas and probability functions for sums of two, three and four independent uniformly distributed random variables are derived.

scholarly journals Characteristics of distribution of amounts of several uniformly distributed random values of information system query treatment times

2019 ◽  
pp. 59-63
Author(s):  
M. M. Butaev
Keyword(s):  
Information Systems ◽  
Random Variables ◽  
Initial Approximation ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Sequential Processing ◽  
Processing Times ◽  
Distribution Laws ◽  
Positive Axis ◽  
Interval Distribution

The normal distribution of a random variable is usually used in studies of the probabilistic characteristics of information systems. However, its use to approximate distributions defined on a limited interval distorts the physical meaning of the model and the numerical results, so it can only be used as an initial approximation. The purpose of the work is the improvement of calculation methods the probabilistic characteristics of information systems. The object of the research is an analytical method for calculating the processing time of a query in the system, the subject is a formula for calculating the duration of a sequential processing of a query by system elements with uniformly distributed random processing times. When deriving formulas for calculating the probability characteristics of a sum of independent uniformly distributed random variables, methods of probability theory are used. For random variables determined only on the positive axis of coordinates, it is proposed to use finite-interval distribution laws, for example, beta distribution. Formulas probability density function and cumulative distribution function for sums of two, three, and four independent uniformly distributed random variables are derived.

scholarly journals Approximation of the difference of two Poisson-like counts by Skellam

2018 ◽  
Vol 55 (2) ◽  
pp. 416-430 ◽  
Author(s):  
H. L. Gan ◽  
Eric D. Kolaczyk
Keyword(s):  
Image Processing ◽  
Network Analysis ◽  
Data Processing ◽  
Count Data ◽  
Random Variables ◽  
Random Variable ◽  
Weak Dependence ◽  
Potential Impact ◽  
The Difference ◽  
Skellam Distribution

AbstractPoisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.

scholarly journals On The Quotient of a Centralized and a Non-centralized Complex Gaussian Random Variable

2020 ◽  
Vol 125 ◽  
Author(s):  
Dazhen Gu
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Complex Signal ◽  
Tolerance Intervals ◽  
Practical Applications ◽  
Second Moments ◽  
Complex Gaussian Random Variable ◽  
The Mean ◽  
Theoretical Developments ◽  
Straightforward Extension

A detailed investigation of the quotient of two independent complex random variables is presented. The numerator has a zero mean, and the denominator has a non-zero mean. A normalization step is taken prior to the theoretical developments in order to simplify the formulation. Next, an indirect approach is taken to derive the statistics of the modulus and phase angle of the quotient. That in turn enables a straightforward extension of the statistical results to real and imaginary parts. After the normalization procedure, the probability density function of the quotient is found as a function of only the mean of the random variable that corresponds to the denominator term. Asymptotic analysis shows that the quotient closely resembles a normally-distributed complex random variable as the mean becomes large. In addition, the first and second moments, as well as the approximate of the second moment of the clipped random variable, are derived, which are closely related to practical applications in complex-signal processing such as microwave metrology of scattering-parameters. Tolerance intervals associated with the ratio of complex random variables are presented.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals An Optimal Tail Selection in Risk Measurement

Risks ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 70
Author(s):  
Małgorzata Just ◽  
Krzysztof Echaust
Keyword(s):  
Mean Squared Error ◽  
Real Data ◽  
Random Variable ◽  
Threshold Level ◽  
Middle Part ◽  
Risk Measurement ◽  
Practical Applications ◽  
Tuning Parameters ◽  
Path Stability ◽  
Asymptotic Mean Squared Error

The appropriate choice of a threshold level, which separates the tails of the probability distribution of a random variable from its middle part, is considered to be a very complex and challenging task. This paper provides an empirical study on various methods of the optimal tail selection in risk measurement. The results indicate which method may be useful in practice for investors and financial and regulatory institutions. Some methods that perform well in simulation studies, based on theoretical distributions, may not perform well when real data are in use. We analyze twelve methods with different parameters for forty-eight world indices using returns from the period of 2000–Q1 2020 and four sub-periods. The research objective is to compare the methods and to identify those which can be recognized as useful in risk measurement. The results suggest that only four tail selection methods, i.e., the Path Stability algorithm, the minimization of the Asymptotic Mean Squared Error approach, the automated Eyeball method with carefully selected tuning parameters and the Hall single bootstrap procedure may be useful in practical applications.

INFORMATION AND UNCERTAINTY IN A QUEUING SYSTEM

2007 ◽  
Vol 21 (3) ◽  
pp. 361-380 ◽  
Author(s):  
Refael Hassin
Keyword(s):  
Service Quality ◽  
Random Variables ◽  
Random Variable ◽  
Queuing System ◽  
Service Rate ◽  
Single Server ◽  
Profit Function

This article deals with the effect of information and uncertainty on profits in an unobservable single-server queuing system. We consider scenarios in which the service rate, the service quality, or the waiting conditions are random variables that are known to the server but not to the customers. We ask whether the server is motivated to reveal these parameters. We investigate the structure of the profit function and its sensitivity to the variance of the random variable. We consider and compare variations of the model according to whether the server can modify the service price after observing the realization of the random variable.

The central limit problem for trimmed sums

1987 ◽  
Vol 102 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Philip S. Griffin ◽  
William E. Pruitt
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Random Variable ◽  
Limit Problem ◽  
Absolute Value ◽  
Trimmed Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Central Limit Problem ◽  
Trimmed Sum

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

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