scholarly journals On the number of empty cells in the allocation scheme of indistinguishable particles

2020 ◽  
Vol 74 (1) ◽  
pp. 15
Author(s):  
Alexey Chuprunov ◽  
Istvan Fazekas
Keyword(s):  
Signal Processing ◽  
Poisson Distribution ◽  
Gaussian Distribution ◽  
Random Variable ◽  
Mathematical Statistics ◽  
Allocation Scheme ◽  
K Cells ◽  
Run Test

The allocation scheme of \(n\) indistinguishable particles into \(N\) different cells is studied. Let the random variable \(\mu_0(n,K,N)\) be the number of empty cells among the first \(K\) cells. Let \(p=\frac{n}{n+N}\). It is proved that \(\frac{\mu_0(n,K,N)-K(1-p)}{\sqrt{ K p(1-p)}}\) converges in distribution to the Gaussian distribution with expectation zero and variance one, when \(n,K, N\to\infty\) such that \(\frac{n}{N}\to\infty\) and \(\frac{n}{NK}\to 0\). If \(n,K, N\to\infty\) so that \(\frac{n}{N}\to\infty\) and \(\frac{NK}{n}\to \lambda\), where \(0<\lambda<\infty\), then \(\mu_0(n,K,N)\) converges in distribution to the Poisson distribution with parameter \(\lambda\). Two applications of the results are given to mathematical statistics. First, a method  is offered to test the value of \(n\). Then, an analogue of the run-test is suggested with an application in signal processing.

scholarly journals Effect of zno nanoparticles on diameter of bubbfil PVA/ZnO nanofibers

2015 ◽  
Vol 19 (4) ◽  
pp. 1447-1449
Author(s):  
Cui-Juan Ning ◽  
Hua Liu ◽  
Na Si ◽  
Ji-Huan He
Keyword(s):  
Size Distribution ◽  
Poisson Distribution ◽  
Gaussian Distribution ◽  
Zno Nanoparticles ◽  
Nano Zno ◽  
Fiber Size ◽  
Zno Nanofibers

The PVA/ZnO nanofibers are obtained by the bubbfil spinning. Distribution of fiber size is tenable by nano-ZnO concentration. Experiment reveals fiber size distribution changes from Gaussian distribution to Poisson distribution when ZnO concentration varies gradually from 2 wt.% to 15 wt.%.

scholarly journals An Alternative Proof For the Minimum Fisher Information of Gaussian Distribution

2018 ◽  
Vol 14 (2) ◽  
pp. 5-10
Author(s):  
Abbas Pak
Keyword(s):  
Signal Processing ◽  
Fisher Information ◽  
Gaussian Distribution ◽  
Estimation Theory ◽  
Alternative Proof ◽  
Statistical Reasoning ◽  
Finite Variance ◽  
Complex Signals

Abstract Fisher information is of key importance in estimation theory. It is used as a tool for characterizing complex signals or systems, with applications, e.g. in biology, geophysics and signal processing. The problem of minimizing Fisher information in a set of distributions has been studied by many researchers. In this paper, based on some rather simple statistical reasoning, we provide an alternative proof for the fact that Gaussian distribution with finite variance minimizes the Fisher information over all distributions with the same variance.

scholarly journals Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula

2008 ◽  
Vol 40 (01) ◽  
pp. 122-143 ◽  
Author(s):  
A. J. E. M. Janssen ◽  
J. S. H. van Leeuwaarden ◽  
B. Zwart
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Poisson Distribution ◽  
Integral Transformation ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Normal Distribution Function ◽  
Gaussian Form ◽  
The Central Limit Theorem ◽  
Leading Term

This paper presents new Gaussian approximations for the cumulative distribution function P(A λ ≤ s) of a Poisson random variable A λ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasi-Gaussian form, which permits evaluation in terms of the normal distribution function Φ. The quasi-Gaussian form contains an implicitly defined function y, which is closely related to the Lambert W-function. A detailed analysis of y leads to a powerful asymptotic expansion and sharp bounds on P(A λ ≤ s). The results for P(A λ ≤ s) differ from most classical results related to the central limit theorem in that the leading term Φ(β), with is replaced by Φ(α), where α is a simple function of s that converges to β as s tends to ∞. Changing β into α turns out to increase precision for small and moderately large values of s. The results for P(A λ ≤ s) lead to similar results related to the Erlang B formula. The asymptotic expansion for Erlang's B is shown to give rise to accurate approximations; the obtained bounds seem to be the sharpest in the literature thus far.

Some characterizations for the Poisson distribution starting with a power-series distribution

1972 ◽  
Vol 71 (3) ◽  
pp. 517-522 ◽  
Author(s):  
D. N. Shanbhag ◽  
R. M. Clark
Keyword(s):  
Power Series ◽  
Poisson Distribution ◽  
Random Variable ◽  
Discrete Random Variable ◽  
Power Series Distribution ◽  
Destructive Process ◽  
Image Position

Let X be a non-negative discrete random variable with distribution {Px} and Y be a random variable denoting the undestroyed part of the random variable X when it is subjected to a destructive process such that

scholarly journals Theorizing Probability Distribution in Applied Statistics

2020 ◽  
Vol 1 (1) ◽  
pp. 79-95
Author(s):  
Indra Malakar
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Poisson Distribution ◽  
Binomial Distribution ◽  
Identical Condition ◽  
Random Variable ◽  
Fixed Number ◽  
Theoretical Knowledge ◽  
Applied Statistics ◽  
Discrete Random Variable

This paper investigates into theoretical knowledge on probability distribution and the application of binomial, poisson and normal distribution. Binomial distribution is widely used discrete random variable when the trails are repeated under identical condition for fixed number of times and when there are only two possible outcomes whereas poisson distribution is for discrete random variable for which the probability of occurrence of an event is small and the total number of possible cases is very large and normal distribution is limiting form of binomial distribution and used when the number of cases is infinitely large and probabilities of success and failure is almost equal.

Discrete Probability Theory

2019 ◽  
pp. 16-42
Author(s):  
Robert H. Swendsen
Keyword(s):  
Probability Theory ◽  
Gaussian Distribution ◽  
Random Variables ◽  
Configurational Entropy ◽  
Random Variable ◽  
Delta Function ◽  
Discrete Probability ◽  
Fundamental Assumption ◽  
Kronecker Delta ◽  
Model Probability

The chapter presents an overview of various interpretations of probability. It introduces a ‘model probability,’ which assumes that all microscopic states that are essentially alike have the same probability in equilibrium. A justification for this fundamental assumption is provided. The basic definitions used in discrete probability theory are introduced, along with examples of their application. One such example, which illustrates how a random variable is derived from other random variables, demonstrates the use of the Kronecker delta function. The chapter further derives the binomial and multinomial distributions, which will be important in the following chapter on the configurational entropy, along with the useful approximation developed by Stirling and its variations. The Gaussian distribution is presented in detail, as it will be very important throughout the book.

scholarly journals Gaussian expansions and bounds for the Poisson distribution applied to the Erlang B formula

2008 ◽  
Vol 40 (1) ◽  
pp. 122-143 ◽  
Author(s):  
A. J. E. M. Janssen ◽  
J. S. H. van Leeuwaarden ◽  
B. Zwart
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Poisson Distribution ◽  
Integral Transformation ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Normal Distribution Function ◽  
Gaussian Form ◽  
The Central Limit Theorem ◽  
Leading Term

This paper presents new Gaussian approximations for the cumulative distribution function P(Aλ ≤ s) of a Poisson random variable Aλ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasi-Gaussian form, which permits evaluation in terms of the normal distribution function Φ. The quasi-Gaussian form contains an implicitly defined function y, which is closely related to the Lambert W-function. A detailed analysis of y leads to a powerful asymptotic expansion and sharp bounds on P(Aλ ≤ s). The results for P(Aλ ≤ s) differ from most classical results related to the central limit theorem in that the leading term Φ(β), with is replaced by Φ(α), where α is a simple function of s that converges to β as s tends to ∞. Changing β into α turns out to increase precision for small and moderately large values of s. The results for P(Aλ ≤ s) lead to similar results related to the Erlang B formula. The asymptotic expansion for Erlang's B is shown to give rise to accurate approximations; the obtained bounds seem to be the sharpest in the literature thus far.

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