Linear Dependence between Two Bernoulli Random Variables

2008 ◽  
Keyword(s):  
Linear Dependence ◽  
Random Variables ◽  
Bernoulli Random Variables

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

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
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.

scholarly journals Distribution of Geometrically Weighted Sum of Bernoulli Random Variables

2011 ◽  
Vol 02 (11) ◽  
pp. 1382-1386 ◽  
Author(s):  
Deepesh Bhati ◽  
Phazamile Kgosi ◽  
Ranganath Narayanacharya Rattihalli
Keyword(s):  
Random Variables ◽  
Weighted Sum ◽  
Bernoulli Random Variables

Comparing sums of exchangeable Bernoulli random variables

1996 ◽  
Vol 33 (02) ◽  
pp. 285-310 ◽  
Author(s):  
Claude Lefèvre ◽  
Sergey Utev
Keyword(s):  
Iterative Procedure ◽  
First Passage Time ◽  
Random Variables ◽  
Passage Time ◽  
Partial Sums ◽  
First Passage ◽  
Sampling Schemes ◽  
Bernoulli Random Variables ◽  
Discrete Random Variables ◽  
Comparison Results

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,· ··, n}.

scholarly journals A refinement of normal approximation to Poisson binomial

2005 ◽  
Vol 2005 (5) ◽  
pp. 717-728 ◽  
Author(s):  
K. Neammanee
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Standard Normal Distribution ◽  
Bernoulli Random Variables ◽  
Binomial Random Variable ◽  
Standard Normal ◽  
Correction Terms ◽  
Better Than

LetX1,X2,…,Xnbe independent Bernoulli random variables withP(Xj=1)=1−P(Xj=0)=pjand letSn:=X1+X2+⋯+Xn.Snis called a Poisson binomial random variable and it is well known that the distribution of a Poisson binomial random variable can be approximated by the standard normal distribution. In this paper, we use Taylor's formula to improve the approximation by adding some correction terms. Our result is better than before and is of order1/nin the casep1=p2=⋯=pn.

The stochastic precedence ordering with applications in sampling and testing

2004 ◽  
Vol 41 (01) ◽  
pp. 73-82 ◽  
Author(s):  
Philip J. Boland ◽  
Harshinder Singh ◽  
Bojan Cukic
Keyword(s):  
Random Sampling ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Simple Random Sampling ◽  
Order Type ◽  
Bernoulli Random Variables ◽  
Binomial Distributions ◽  
Necessary And Sufficient ◽  
Stochastic Precedence

Stratified and simple random sampling (or testing) are two common methods used to investigate the number or proportion of items in a population with a particular attribute. Although it is known that cost factors and information about the strata in the population are often crucial in deciding whether to use stratified or simple random sampling in a given situation, the stochastic precedence ordering for random variables can also provide the basis for an interesting criteria under which these methods may be compared. It may be particularly relevant when we are trying to find as many special items as possible in a population (for example individuals with a disease in a country). Properties of this total stochastic order on the class of random variables are discussed, and necessary and sufficient conditions are established which allow the comparison of the number of items of interest found in stratified random sampling with the number found in simple random sampling in the stochastic precedence order. These conditions are compared with other results established on stratified and simple random sampling (testing) using different stochastic-order-type criteria, and applications are given for the comparison of sums of Bernoulli random variables and binomial distributions.

scholarly journals On the Convolution of Heterogeneous Bernoulli Random Variables

2011 ◽  
Vol 48 (3) ◽  
pp. 877-884 ◽  
Author(s):  
Maochao Xu ◽  
N. Balakrishnan
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Random Variables ◽  
Likelihood Ratio Order ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Bernoulli Random Variables

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).

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