Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants

Abstract Given random variables X and Y having finite moments of all orders, their uncorrelatedness set is defined as the set of all pairs (j, k) ∈ ℕ2, for which Xj and Yk are uncorrelated. It is known that, broadly put, any subset of ℕ2 can serve as an uncorrelatedness set. This claim is no longer valid for random variables with prescribed distributions, in which case the need arises so as to identify the possible uncorrelatedness sets. This paper studies the uncorrelatedness sets for positive random variables uniformly distributed on three points. Some general features of these sets are derived. Two related Vandermonde-type determinants are examined and applied to describe uncorrelatedness sets in some special cases.

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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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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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On Exact Sampling of Nonnegative Infinitely Divisible Random Variables

Advances in Applied Probability ◽

10.1239/aap/1346955267 ◽

2012 ◽

Vol 44 (3) ◽

pp. 842-873 ◽

Cited By ~ 3

Author(s):

Zhiyi Chi

Keyword(s):

Basic Result ◽

Random Variables ◽

Random Variable ◽

Fixed Number ◽

Rejection Sampling ◽

Infinitely Divisible ◽

Exact Sampling ◽

Special Cases ◽

Wide Range ◽

Lévy Density

Nonnegative infinitely divisible (i.d.) random variables form an important class of random variables. However, when this type of random variable is specified via Lévy densities that have infinite integrals on (0, ∞), except for some special cases, exact sampling is unknown. We present a method that can sample a rather wide range of such i.d. random variables. A basic result is that, for any nonnegative i.d. random variable X with its Lévy density explicitly specified, if its distribution conditional on X ≤ r can be sampled exactly, where r > 0 is any fixed number, then X can be sampled exactly using rejection sampling, without knowing the explicit expression of the density of X. We show that variations of the result can be used to sample various nonnegative i.d. random variables.

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