Binomial Approximation to a Poisson Random Variable

2007 ◽  
Keyword(s):  
Random Variable ◽  
Poisson Random Variable ◽  
Binomial Approximation

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.

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.

On the limiting behaviour of a non-homogeneous Markovian manpower model with independent Poisson input

1982 ◽  
Vol 19 (2) ◽  
pp. 433-438 ◽  
Author(s):  
P.-C. G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Markov Chain Model ◽  
Random Variable ◽  
Homogeneous Markov Chain ◽  
Chain Model ◽  
Poisson Random Variable ◽  
Poisson Input ◽  
Manpower System ◽  
Limiting Behaviour ◽  
Mutually Independent

We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually independent Poisson variates.

Poisson approximation for (k1, k2)-events via the Stein-Chen method

2004 ◽  
Vol 41 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
P. Vellaisamy
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Success Probability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Poisson Random Variable ◽  
Markov Dependent Trials ◽  
Special Case

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k1, k2) denote the number of times that k1 failures are followed by k2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k1, k2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for Nk3(n; k1, k2), the number of times that k1 failures followed by k2 successes occur k3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

Poisson approximations for runs and patterns of rare events

1991 ◽  
Vol 23 (4) ◽  
pp. 851-865 ◽  
Author(s):  
Anant P. Godbole
Keyword(s):  
Success Probability ◽  
Standard Condition ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Poisson Random Variable ◽  
Runs And Patterns ◽  
Reliability Systems ◽  
Poisson Approximations

Consider a sequence of Bernoulli trials with success probability p, and let Nn,k denote the number of success runs of length among the first n trials. The Stein–Chen method is employed to obtain a total variation upper bound for the rate of convergence of Nn,k to a Poisson random variable under the standard condition npk→λ. This bound is of the same order, O(p), as the best known for the case k = 1, i.e. for the classical binomial-Poisson approximation. Analogous results are obtained for occurrences of word patterns, where, depending on the nature of the word, the corresponding rate is at most O(pk–m) for some m = 0, 2, ···, k – 1. The technique is adapted for use with two-state Markov chains. Applications to reliability systems and tests for randomness are discussed.

On the limiting behaviour of a non-homogeneous Markovian manpower model with independent Poisson input

1982 ◽  
Vol 19 (02) ◽  
pp. 433-438 ◽  
Author(s):  
P.-C. G. Vassiliou
Keyword(s):  
Markov Chain ◽  
Markov Chain Model ◽  
Random Variable ◽  
Homogeneous Markov Chain ◽  
Chain Model ◽  
Poisson Random Variable ◽  
Poisson Input ◽  
Manpower System ◽  
Limiting Behaviour ◽  
Mutually Independent

We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually independent Poisson variates.

Renewal theory in two dimensions: bounds on the renewal function

1977 ◽  
Vol 9 (03) ◽  
pp. 527-541 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Renewal Theory ◽  
Random Variable ◽  
Two Dimensions ◽  
Upper And Lower Bounds ◽  
Renewal Processes ◽  
Renewal Function ◽  
Poisson Random Variable ◽  
Joint Distributions ◽  
The Family ◽  
Fréchet Bounds

In two earlier papers [6], [7] the properties of bivariate renewal processes and their associated two-dimensional renewal functions, H(x, y) were examined. By utilising the Fréchet bounds for joint distributions and the properties of univariate renewal processes, a collection of upper and lower bounds for H(x, y) are constructed. The evaluation of these bounds is carried out for the case of the family of bivariate Poisson processes. An interesting by-product of this investigation leads to a new inequality for the median of a Poisson random variable.

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

2002 ◽  
Vol 34 (3) ◽  
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 variables X1, X2, …, Xn are said to be totally negatively dependent (TND) if and only if the random variables Xi and ∑j≠iXj are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X1, x2, …, Xn with P[Xi = 1] = pi = 1 - P[Xi = 0], an upper bound for the total variation distance between ∑ni=1Xi and 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.

Uniform saddlepoint approximations

1988 ◽  
Vol 20 (3) ◽  
pp. 622-634 ◽  
Author(s):  
J. L. Jensen
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Tail Probability ◽  
Poisson Random Variable ◽  
Compound Poisson ◽  
Saddlepoint Approximations ◽  
The Mean ◽  
General Conditions

The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to ∞. This is done for the expansions of the density and of the tail probability of the mean of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum , where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X1 introduced in Daniels (1954).

Poisson approximation for some statistics based on exchangeable trials

1983 ◽  
Vol 15 (3) ◽  
pp. 585-600 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Total Variation ◽  
Limit Theorems ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Variation Distance ◽  
Image Position ◽  
Poisson Approximations

Stein's (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables.Let be exchangeable random elements of a space and, for I a k-subset of , let XI be a 0–1 function. The statistics studied here are of the form where N is some collection of k -subsets of .An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.

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