A representation for discrete distributions by equiprobable mixtures

We obtain a representation of an arbitrary discrete distribution with n mass points by an equiprobable mixture of r distributions, each of which has no more than a (≧2) mass points, where r is the smallest integer greater than or equal to (n – 1)/(a – 1). An application to the generation of discrete random variables on a computer is described, which has as an important special case Walker's (1977) alias method.

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Sufficient Conditions for Some Stochastic Orders of Discrete Random Variables with Applications in Reliability

Mathematics ◽

10.3390/math10010147 ◽

2022 ◽

Vol 10 (1) ◽

pp. 147

Author(s):

Félix Belzunce ◽

Carolina Martínez-Riquelme ◽

Magdalena Pereda

Keyword(s):

Likelihood Ratio ◽

Sufficient Conditions ◽

Random Variables ◽

Discrete Distributions ◽

Stochastic Orders ◽

Survival Functions ◽

Discrete Random Variables ◽

Parametric Families

In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions.

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SOME NEW RELIABILITY BOUNDS FOR SUMS OF NBUE RANDOM VARIABLES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964810000264 ◽

2010 ◽

Vol 25 (1) ◽

pp. 83-102

Author(s):

Steven G. From

Keyword(s):

Lower Bounds ◽

Random Variables ◽

Upper And Lower Bounds ◽

Independent Random Variables ◽

Important Special Case ◽

Lifetime Distributions ◽

Survivor Function ◽

Numerical Studies ◽

Special Case ◽

Better Than

In this article, we discuss some new upper and lower bounds for the survivor function of the sum of n independent random variables each of which has an NBUE (new better than used in expectation) distribution. In some cases, only the means of the random variables are assumed known. These bounds are compared to the sharp bounds given in Cheng and Lam [6], which requires both means and variances known. Although the new bounds are not sharp, they often produce better upper bounds for the survivor function in the extreme right tail of many NBUE lifetime distributions, an important special case in applications. Moreover, a lower bound exists in one case not handled by the lower bounds of Theorem 3 in Cheng and Lam [6]. Numerical studies are presented along with theoretical discussions.

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Normal approximations to discrete unimodal distributions

Journal of Applied Probability ◽

10.1017/s0021900200115931 ◽

1986 ◽

Vol 23 (04) ◽

pp. 1013-1018

Author(s):

B. G. Quinn ◽

H. L. MacGillivray

Keyword(s):

Stationary Distribution ◽

Sufficient Conditions ◽

Random Variables ◽

Hypergeometric Distribution ◽

Death Process ◽

Normal Approximations ◽

Discrete Random Variables ◽

Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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The number of failed components in a coherent working system when the lifetimes are discretely distributed

Metrika ◽

10.1007/s00184-021-00817-2 ◽

2021 ◽

Author(s):

Krzysztof Jasiński

Keyword(s):

Random Variables ◽

Continuous Case ◽

Coherent System ◽

Coherent Systems ◽

Discrete Random Variables ◽

Independent And Identically Distributed

AbstractIn this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, $$i=0,\ldots ,n-k,$$ i = 0 , … , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature.

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Approximation of continuous random variables for the evaluation of the reliability parameter of complex stress–strength models

Annals of Operations Research ◽

10.1007/s10479-021-04010-6 ◽

2021 ◽

Author(s):

Alessandro Barbiero ◽

Asmerilda Hitaj

Keyword(s):

Order Approximation ◽

Linear Optimization ◽

Closed Form Solution ◽

Random Variables ◽

Discrete Distribution ◽

Computation Time ◽

Form Solution ◽

Engineering Problem ◽

Approximation Technique ◽

Reliability Parameter

AbstractIn many management science or economic applications, it is common to represent the key uncertain inputs as continuous random variables. However, when analytic techniques fail to provide a closed-form solution to a problem or when one needs to reduce the computational load, it is often necessary to resort to some problem-specific approximation technique or approximate each given continuous probability distribution by a discrete distribution. Many discretization methods have been proposed so far; in this work, we revise the most popular techniques, highlighting their strengths and weaknesses, and empirically investigate their performance through a comparative study applied to a well-known engineering problem, formulated as a stress–strength model, with the aim of weighting up their feasibility and accuracy in recovering the value of the reliability parameter, also with reference to the number of discrete points. The results overall reward a recently introduced method as the best performer, which derives the discrete approximation as the numerical solution of a constrained non-linear optimization, preserving the first two moments of the original distribution. This method provides more accurate results than an ad-hoc first-order approximation technique. However, it is the most computationally demanding as well and the computation time can get even larger than that required by Monte Carlo approximation if the number of discrete points exceeds a certain threshold.

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On improvements of the order of approximation in the Poisson limit theorem

Journal of Applied Probability ◽

10.1017/s0021900200103808 ◽

1996 ◽

Vol 33 (01) ◽

pp. 146-155 ◽

Cited By ~ 1

Author(s):

K. Borovkov ◽

D. Pfeifer

Keyword(s):

Limit Theorem ◽

Random Variables ◽

Poisson Approximation ◽

Order Of Approximation ◽

Operator Semigroups ◽

Bernoulli Random Variables ◽

Usual Rate ◽

Poisson Limit ◽

Poisson Measures ◽

Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

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ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE

Stochastics and Dynamics ◽

10.1142/s0219493704000900 ◽

2004 ◽

Vol 04 (01) ◽

pp. 63-76 ◽

Cited By ~ 24

Author(s):

OLIVER JENKINSON

Keyword(s):

Hausdorff Dimension ◽

Finite Subset ◽

Continued Fraction ◽

Cantor Set ◽

Computational Method ◽

Accurate Approximation ◽

Natural Numbers ◽

Important Special Case ◽

The One ◽

Special Case

Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim (EA) is between 0 and 1. It is shown that the set [Formula: see text] intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim (EA), and employ it to investigate numerically the way in which [Formula: see text] intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbański, that [Formula: see text] is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim (E{1,2}), improving on the one given in [18].

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Two-stage stochastic programming problems involving interval discrete random variables

OPSEARCH ◽

10.1007/s12597-012-0078-1 ◽

2012 ◽

Vol 49 (3) ◽

pp. 280-298 ◽

Cited By ~ 2

Author(s):

Suresh Kumar Barik ◽

Mahendra Prasad Biswal ◽

Debashish Chakravarty

Keyword(s):

Stochastic Programming ◽

Random Variables ◽

Two Stage ◽

Discrete Random Variables

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An Elementary Proof of Suslin Reciprocity

Canadian Mathematical Bulletin ◽

10.4153/cmb-2005-020-x ◽

2005 ◽

Vol 48 (2) ◽

pp. 221-236 ◽

Cited By ~ 4

Author(s):

Matt Kerr

Keyword(s):

Elementary Proof ◽

Function Fields ◽

Algebraic Cycles ◽

Important Special Case ◽

Special Case

AbstractWe state and prove an important special case of Suslin reciprocity that has found significant use in the study of algebraic cycles. An introductory account is provided of the regulator and norm maps on Milnor K2-groups (for function fields) employed in the proof.

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