Binomial Distribution Function Research Articles

A uniform bound on negative binomial approximation with w-functions

This paper uses Stein’s method and w-functions to determine a uniform bound for the Kolmogorov distance between the cumulative distribution function of a non-negative integer-valued random variable X and the negative binomial cumulative distribution function with parameters 0 r > and 1( 0,1) pq =− ∈ , where () rq p EX = and () EX is the mean of X. Two examples are provided to

Bayesian inference of cosmic density fields from non-linear, scale-dependent, and stochastic biased tracers

We present a Bayesian reconstruction algorithm to generate unbiased samples of the underlying dark matter field from halo catalogues. Our new contribution consists of implementing a non-Poisson likelihood including a deterministic non-linear and scale-dependent bias. In particular we present the Hamiltonian equations of motions for the negative binomial (NB) probability distribution function. This permits us to efficiently sample the posterior distribution function of density fields given a sample of galaxies using the Hamiltonian Monte Carlo technique implemented in the Argo code. We have tested our algorithm with the Bolshoi $N$-body simulation at redshift $z = 0$, inferring the underlying dark matter density field from sub-samples of the halo catalogue with biases smaller and larger than one. Our method shows that we can draw closely unbiased samples (compatible within 1-$\sigma$) from the posterior distribution up to scales of about $k$~1 h/Mpc in terms of power-spectra and cell-to-cell correlations. We find that a Poisson likelihood yields reconstructions with power spectra deviating more than 10% at $k$=0.2 h/Mpc. Our reconstruction algorithm is especially suited for emission line galaxy data for which a complex non-linear stochastic biasing treatment beyond Poissonity becomes indispensable.

Arrival modelling for molecular communication via diffusion

The arrival of molecules in molecular communication via diffusion obeys, by its nature, the binomial distribution, considering the hitting probability as the success probability. It is, however, hard to work with the binomial cumulative distribution function (CDF) when considering consecutively sent symbols as it is necessary to add the distribution. Therefore, in the literature, two approximations of the binomial distribution are used. In the present reported work, the Gaussian and Poisson approximations of the binomial model of the molecule arrival process have been analysed. Considering the distance and the number of emitted molecules, the regions in which the Poisson or Gaussian model is better in terms of root mean squared error of CDFs are investigated and the regions using numerical simulations are confirmed.

ON BINOMIAL APPROXIMATION FOR INDEPENDENT BERNOULLI RANDOM SUMS

A non-uniform bound for the distance between the distribution function of random sums of independent Bernoulli random variables and an appropriate binomial distribution function could be obtained. Two examples have been given to illustrate the result obtained.

ON NEGATIVE BINOMIAL APPROXIMATION FOR GEOMETRIC RANDOM SUMS

We give a non-uniform bound for the distance between the dis- tribution function of random sums of independent geometric random variables and an appropriate negative binomial distribution function. Two examples have been given to illustrate the result obtained.

On Narrowing Chain-length Distributions in Ideally Dispersed Polymerization Systems

Theoretical considerations and Monte Carlo simulations prove that small size of droplets narrows dispersity of chains in both chain and step polymerizations. Equations describing the chains lengths distribution in living polymerization carried out in nano-droplets are derived, proving to be the binomial distribution function. The effect of lowering of polymer dispersity in irreversible polycondensation is predicted as well.

Improved Inequalities for the Poisson and Binomial Distribution and Upper Tail Quantile Functions

The exact evaluation of the Poisson and Binomial cumulative distribution and inverse (quantile) functions may be too challenging or unnecessary for some applications, and simpler solutions (typically obtained by applying Normal approximations or exponential inequalities) may be desired in some situations. Although Normal distribution approximations are easy to apply and potentially very accurate, error signs are typically unknown; error signs are typically known for exponential inequalities at the expense of some pessimism. In this paper, recent work describing universal inequalities relating the Normal and Binomial distribution functions is extended to cover the Poisson distribution function; new quantile function inequalities are then obtained for both distributions. Exponential bounds—which improve upon the Chernoff-Hoeffding inequalities by a factor of at least two—are also obtained for both distributions.

A BOUND ON THE POISSON BINOMIAL-POISSON RELATIVE ERROR

We use the Stein-Chen method to determine a bound on the rel- ative error between the Poisson binomial distribution function with parameter p = (p1,...,pn) and the Poisson distribution function with mean � = P n=1 pi 1 pi . With this bound, the Poisson distribution function with this mean can be used as an estimate of the Poisson binomial distribution function whenever all pi are small.

Disparities in Use of Laparoscopic Hysterectomies: A Nationwide Analysis

Study ObjectiveTo determine patient and hospital characteristics that were associated with undergoing laparoscopic hysterectomy compared with abdominal hysterectomy. DesignCanadian Task Force Classification II-3. MethodsIn this retrospective cohort study, we analyzed the 2010 Healthcare Cost and Utilization Project Nationwide Inpatient Sample database. All women who underwent laparoscopic or abdominal hysterectomy for either menorrhagia or leiomyoma were included based on International Classification of Diseases, Ninth Revision coding. A linear model with binomial distribution and logit link function was used to determine patient and hospital characteristics associated with hysterectomy approach. Main ResultsA total of 32 436 patients were included in this study. Of these, 32% patients underwent laparoscopic hysterectomies, and 67% underwent abdominal hysterectomies. With regard to patient characteristics, women younger than 35 years old were more likely to undergo laparoscopic hysterectomy when compared with each of the other age categories (p < .001). White women were more likely to undergo laparoscopic hysterectomy than black women, Hispanic women, or women classified as “other” races (p < .001 for all comparisons). With regard to median income, patients from the lowest national quartile were less likely to undergo laparoscopic hysterectomy when compared with each of the other 3 national quartiles for income (p = .01, p < .001, p = .001, respectively). Payment by private insurance was associated with laparoscopic hysterectomy when compared with payment by Medicare or payment by insurance category “other” (p < .001 for both). With regard to hospital characteristics, hospitals in the Northeast were more likely to have laparoscopic hysterectomies than hospitals in the Midwest or South (p < .001 for both comparisons); urban hospitals were more likely than rural hospitals (p < .001); teaching hospitals were more likely than nonteaching hospitals (p < .001); and government-owned hospitals were less likely than private, nonprofit or private, investor owned (p < .001 for both comparisons). ConclusionsDespite the increased popularity of and training in laparoscopic hysterectomies, there remains an obvious disparity in its delivery with regard to patient and hospital characteristics. Further investigation is needed on the etiology of this disparity and interventions that may alleviate it.

Identifying the Factors that Influence Change in SEBD Using Logistic Regression Analysis

Multiple linear regression and ANOVA models are widely used in applications since they provide effective statistical tools for assessing the relationship between a continuous dependent variable and several predictors. However these models rely heavily on linearity and normality assumptions and they do not accommodate categorical dependent variables. The seminal contribution of John Nelder and Robert Wedderburn (1972) introduced the concept of Generalized Linear Models. GLMs overcome the limitations of Normal regression models and accommodate any distribution which is a member of the exponential family. Moreover, these models relate the dependent variable to the linear predictor (non-random component) through any invertible link function. Logistic regression models are GLMs that accommodate categorical dependent variables. They assume a Binomial distribution and Logit canonical link function. The iteratively re-weighted least squares algorithm using the Fisher scoring technique is employed to maximize the log-likelihood function in GLMs and estimate the model parameters. In this paper, Logistic regression analysis was used to identify the dominant factors that influence change in social, emotional and behaviour difficulties (SEBD) of Maltese children. The study comprised 486 pupils whose SEBD was assessed by both teachers and parents using the Strengths and Difficulties Questionnaire (Goodman 1997) when the children were aged 6 and 9 years old.

Atom probe analysis on interaction between Cr and N in bake-hardening steels with anti-aging properties at RT

One-dimensional atom probe (1DAP) analysis was performed on chromium and nitrogen added bake hardening steel sheets that have both high bake-hardenability and anti-aging properties at room temperature (RT). The atomic data of more than 2 million atoms were collected for sample steels with and without low-temperature aging after recrystallization annealing and quenching. The correlation in atomic position between chromium and nitrogen atoms in a solid solution was investigated by a statistical analysis using the binomial distribution function. In the samples with low-temperature aging, the probability that a chromium atom was observed near a nitrogen atom was significantly higher than that estimated from the null hypothesis that there was no attractive atomic interaction. This suggests that there is an attractive interaction between a nitrogen atom and a chromium atom in bcc iron, which led to the anti-aging properties at RT. In contrast, such correlation was not observed definitely in the samples without low-temperature aging, which implied that the atomic pair formation is a thermal activation process.

APPROXIMATION OF GENERALIZED BINOMIAL BY POISSON DISTRIBUTION FUNCTION

The Stein-Chen method and the w-function associated with gen- eralized binomial random variable are used to determine a non-uniform bound on Poisson approximation to generalized binomial distribution function. Some numerical examples are given to illustrate applications of the result.

Multiplicity distributions in the forward rapidity region in proton-proton collisions at the Large Hadron Collider

Measured multiplicity distributions of primary charged particles produced in the forward rapidity region of the proton-proton ($pp$) collisions at the center-of-mass energy, $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$, at the LHC have been analyzed in terms of the negative binomial distribution function. Like the multiplicity distributions in the midrapidity region for the $pp$ collisions at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$, the distributions for the minimum bias events in the forward region are also better described with the superposition of two negative binomail distributions, as proposed by a two-component model of particle production from two processes, the soft and the hard. However, the multiplicity distribution for the ``hard-QCD'' events in a large pseudorapidity window does not oblige the two-component model.

Efficient stochastic simulation of simultaneous reaction and diffusion in a gas-liquid interface

A stochastic simulation of simultaneous reaction and diffusion is proposed for the gas-liquid interface formed in the surface of a gas bubble within a liquid. The interface between a carbon dioxide bubble and an aqueous solution of calcium hydroxide was simulated as an application example, taken from the integrated production of calcium carbonate. First Gillespie’s stochastic simulation algorithm was applied in separate reaction and diffusion simulations. The results from these simulations were consistent with deterministic solutions based on differential equations. However it was observed that stochastic diffusion simulations are extremely slow. The sampling of diffusion events was accelerated applying a group molecule transfer scheme based on the binomial distribution function. Simulations of the reaction-diffusion in the gas-liquid interface based on the standard Gillespie’s stochastic algorithm were also slow. However the application of the binomial distribution function scheme allowed to compute the concentration profiles in the gas-liquid interface in a fraction of the time required with the standard Gillespie’s stochastic algorithm.

AN IMPROVED BOUND ON THE POISSON-NEGATIVE BINOMIAL ERROR

The Stein-Chen method is used to obtain a new non-uniform bound on the error of the negative binomial cumulative distribution function with parameters n and p and the Poisson cumulative distribution function with mean n(1−p). The bound obtained in this study is sharper than those reported in (8).

A three-component description of multiplicity distributions in pp collisions at the LHC

A possible signal of new phenomena emerging in the global characteristics of multiparticle production in hadron interactions at TeV energies is studied. The multiplicity distributions of charged particles measured in proton–proton collisions at the Large Hadron Collider at CERN are analyzed in the phenomenological framework of the weighted superposition of three negative binomial distribution functions. Examination of the experimental data indicates the existence of a narrow peak at low multiplicities which can be described by a separate component in the total distribution. The multiplicity characteristics of the third component reveal approximate energy and pseudorapidity invariance, of which a physical explanation represents a challenging problem in high energy multiple particle production.

Reliability Analysis of Quality Inspection Based on Bernoulli Probability Mode Binomial Distribution Function

Reliability Analysis of Quality Inspection Based on Bernoulli Probability Mode Binomial Distribution Function

Theoretical derivation of precipitation coverage probability from a set of heterogeneous point probabilities

Various meteorological studies require probabilistic modeling such as the flood, drought, and wet and dry spell occurrences. The most commonly employed approaches are based on the binomial probability distribution function (pdf), which is valid for homogeneous temporal and spatial probabilities. However, in practice, heterogeneous probabilities are very common, and in this case, the classical binomial pdf cannot be employed. This paper presents a rigorous probabilistic methodology for heterogeneous probabilities by considerations from the random field concept. The general form of the binomial pdf is derived and it is shown that for homogeneous probabilities it yields the classical binomial pdf exactly. The necessary analytical formulations are derived with the assumption that the meteorological occurrences at a set of sites or time periods (day, month, and year) are independent from each other. The probability statements derived in this paper are helpful in predicting the regional and temporal probable potential future meteorological occurrences. The application of the proposed methodology is presented for annual precipitation occurrences at six meteorology stations around Istanbul, Turkey.

THE LEAST UPPER BOUND ON THE POISSON-BINOMIAL RELATIVE ERROR

A simple mathematical method is used to determine the least up- per bound on the relative error between the binomial cumulative distribution function with parameters n and p and the Poisson cumulative distribution func- tion with mean � = np q = np 1 p . With this upper bound, the Poisson cumulative distribution function with this mean can be used as an estimate of the binomial cumulative distribution function when p is sufficiently small even ifis rather large. By numerical comparison, the upper bound obtained in this study is sharper than those reported in Teerapabolarn (4).

Prediction of the types of ion channel-targeted conotoxins based on radial basis function network

Conotoxins are small disulfide-rich peptide toxins, which have the exceptional diversity of sequences. Because conotoxins are able to specifically bind to ion channels and interfere with neurotransmission, they are considered as the excellent pharmacological candidates in drug design. Appropriate type assignment of newly sequenced mature ion channel-targeted conotoxins with computational method is conducive to explore the biological and pharmacological functions of conotoxins. In this paper, we developed a novel method based on binomial distribution and radial basis function network to predict the types of ion-channel targeted conotoxins. We achieved the overall accuracy of 89.3% with average accuracy of 89.7% in the prediction of three types of ion channel-targeted conotoxins in jackknife cross-validation test, indicating that the method is superior to other state-of-the-art methods. In addition, we evaluated the proposed model with an independent dataset including 77 conotoxins. The overall accuracy of 85.7% was achieved, validating that our model is reliable. Moreover, we used the proposed method to annotate 336 function-undefined mature conotoxins in the UniProt Database. The model provides the valuable instructions for theoretical and experimental research on conotoxins.