Two Methods for the Expectation in Envelopes Pairing Problem

2015 ◽  
Vol 713-715 ◽  
pp. 2016-2019
Author(s):  
Lei Qin ◽  
Kang Zhou
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Mathematical Induction ◽  
General Situation ◽  
Distribution Law ◽  
Pairing Problem

This paper introduces two methods for solving the expectation in Envelopes Pairing Problem, one is that the random variable is expressed as the sum of n random variables and then solves its expectation; another method is put forward by this paper, that is, firstly the distribution law is derived from simple cases, then considers problems of general situation, gets the expectation of conjecture by simple cases, finally proves the conjecture with mathematical induction.

On Random Variables which have the same Distribution as their Reciprocals

1965 ◽  
Vol 8 (6) ◽  
pp. 819-824 ◽  
Author(s):  
V. Seshadri
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Interesting Property ◽  
Distribution Law ◽  
Remarkable Property ◽  
F Distribution

The motivation for this paper lies in the following remarkable property of certain probability distributions. The distribution law of the r. v. (random variable) X is exactly the same as that of 1/ X, and in the case of a r. v. with p. d. f. (probability density function) f(x; a, b) where a, b are parameters, the p. d. f. of 1/X is f(x; b, a). In the latter case the p. d. f. of the reciprocal is obtained from the p. d. f. of X by merely switching the parameters. The existence of random variables with this property is perhaps familiar to statisticians, as is evidenced by the use of the classical 'F' distribution. The Cauchy law is yet another example which illustrates this property. It seems, therefore, reasonable to characterize this class of random variables by means of this rather interesting property.

MODEL OF DIGITAL FIBER-OPTICAL MEASUREMENT SYSTEM

2019 ◽  
Author(s):  
Sergii Okocha ◽  
Andrew Petrenko
Keyword(s):  
Fiber Optic ◽  
Optical Measurement ◽  
Random Variables ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Independent Random Variables ◽  
Distribution Law ◽  
Useful Signal ◽  
Accuracy Of Measurements ◽  
Measuring Signal

A new approach is proposed to obtain a generalized model of distribu­ -ted digital fiber-optic measuring systems of interferometric type using multichannel reception of signals of a fiber-optic inter-mode interferometer to improve the accuracy of measurements. On the basis of this approach, generalized equations for the con-version of fiber-to-digital converters of the geometric coordinates of the points of the measured object are obtained. The equations combine all the private mathema­ ti­ cal models of energy information processes. The approach is based on the representa-tion of the "coordinate of point (move) — code" in the form of an equation of perfect digital-to-analog source code conversion, the processes of which change bit codes are given in the form of logical functions from the input move and points of real multidimensional spatial parameters. The fiber optic line is used in bidirectional optical sig-nal mode in conjunction with the code element element. In this function, the supply of radiation from the measuring units to the points of reading information, the control ele­ -ment, transmitters of modulated radiation are combined in a single fiber. The spatial separation of optical streams is carried out in a block of bidirectional optical communication devices, which is a set of fiber-optic Y-splitters. For multichannel reception, the principle of making a decision on registration of influence on the interferometer is in-troduced: if the module of the output signal exceeds the set level, the signal is fixed. Changes in the measuring signal from external conditions are determined by changes in the parameters of the fiber, the processes of interaction of modes and double re-fraction. Changes in the measurement signal are presented as random variables. Using the central limit theorem for a large number of double sums, the values of the signals at a particular point in time are described by independent random variables, with a normal distribution law and a variance. The beneficial effect is considered regu-lar, and at the time of measurement it is represented by a centered Gaussian random variable with variance. The useful signal component is a Gaussian random variable with standard deviation.

ANALYSIS OF RELIABILITY OF FLAT TROUSERS BASED ON P-BLOCKS

2020 ◽  
Author(s):  
Anastasiya Solov'eva ◽  
Sergey Solov'ev ◽  
Leonid Shevcov ◽  
Valeriya Piven'
Keyword(s):  
Random Variables ◽  
Physical And Mechanical Properties ◽  
Random Variable ◽  
Statistical Information ◽  
Distribution Law ◽  
Coating Materials ◽  
Lack Of Information ◽  
Stochastic Parameters ◽  
Pavement Structures

The paper considers an approach to the probabilistic analysis of the reliability of flat trusses based on p-boxes (probability boxes, p-boxes). Modeling of stochastic parameters in the form of p-blocks is justified for building pavement structures due to significant variability of climatic loads, variations in the physical and mechanical properties of coating materials, installation tolerances and other uncertainties. The advantage of this method is the possibility of using it with incomplete (limited) statistical information - when it is difficult to determine the probability distribution law or the parameters of a random variable. Variants of constructing p-blocks are illustrated for various types of incompleteness of statistical information: for an unknown distribution law using Chebyshev's inequality, for interval estimates of the parameters of random variables, etc. Information is given on the possibility of performing algebraic operations on p-blocks. The probability of no-failure operation with such approaches will be presented as an interval of values. If the interval is too wide (uninformative), the quality of statistical information should be improved by conducting additional tests. The paper presents mathematical models of limiting states taking into account the variability of the basic random variables. The possibility of using the proposed approach in the framework of most practical problems in the construction industry for assessing the safety of statically definable farms is shown. As a result, a formula is given for assessing the reliability of a truss as a conditional mechanical sequential system (in terms of the theory of reliability), taking into account the lack of information about the dependence of its elements. The algorithm for analyzing reliability is considered on a numerical example. The developed approach can be used for other types of statically definable hinge-rod systems.

scholarly journals Bayes' Model of the Best-Choice Problem with Disorder

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
Vladimir Mazalov ◽  
Evgeny Ivashko
Keyword(s):  
Decision Maker ◽  
Random Variables ◽  
Random Variable ◽  
Random Time ◽  
Choice Problem ◽  
Distribution Law ◽  
Optimal Rule ◽  
Bayes Model ◽  
Best Choice Problem

We consider the best-choice problem with disorder and imperfect observation. The decision-maker observes sequentially a known number of i.i.d random variables from a known distribution with the object of choosing the largest. At the random time the distribution law of observations is changed. The random variables cannot be perfectly observed. Each time a random variable is sampled the decision-maker is informed only whether it is greater than or less than some level specified by him. The decision-maker can choose at most one of the observation. The optimal rule is derived in the class of Bayes' strategies.

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.

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.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

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