Approximating gamma distributions by normalized negative binomial distributions

1994 ◽  
Vol 31 (02) ◽  
pp. 391-400 ◽  
Author(s):  
José A. Adell ◽  
Jesús De La Cal
Keyword(s):  
Distribution Function ◽  
Negative Binomial ◽  
Upper And Lower Bounds ◽  
Exact Order ◽  
Bernstein Type ◽  
Birth Process ◽  
Charged Multiplicity ◽  
Binomial Distributions ◽  
Gamma Distributions ◽  
Multiplicity Distributions

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p , where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

Approximating gamma distributions by normalized negative binomial distributions

1994 ◽  
Vol 31 (2) ◽  
pp. 391-400 ◽  
Author(s):  
José A. Adell ◽  
Jesús De La Cal
Keyword(s):  
Distribution Function ◽  
Negative Binomial ◽  
Upper And Lower Bounds ◽  
Exact Order ◽  
Bernstein Type ◽  
Birth Process ◽  
Charged Multiplicity ◽  
Binomial Distributions ◽  
Gamma Distributions ◽  
Multiplicity Distributions

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p, where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

scholarly journals INFLUENCE OF STATISTICAL FLUCTUATIONS ON K/π RATIOS IN RELATIVISTIC HEAVY ION COLLISIONS

2001 ◽  
Vol 16 (07) ◽  
pp. 1227-1235 ◽  
Author(s):  
C. B. YANG ◽  
X. CAI
Keyword(s):  
Negative Binomial ◽  
Heavy Ion ◽  
Relativistic Heavy Ion Collisions ◽  
Relative Variance ◽  
Ion Collisions ◽  
Statistical Fluctuations ◽  
Binomial Distributions ◽  
The Mean ◽  
Multiplicity Distributions ◽  
Statistical Background

The influence of pure statistical fluctuations on K/π ratio is investigated in an event-by-event way. Poisson and the modified negative binomial distributions are used as the multiplicity distributions since they both have statistical background. It is shown that the distributions of the ratio in these cases are Gaussian, and the mean and relative variance are given analytically.

scholarly journals Multiplicity moments using Tsallis statistics in high-energy hadron–nucleus interactions

2020 ◽  
Vol 29 (04) ◽  
pp. 2050021
Author(s):  
S. Sharma ◽  
G. Chaudhary ◽  
K. Sandeep ◽  
A. Singla ◽  
M. Kaur
Keyword(s):  
High Energy ◽  
Fixed Target ◽  
Charged Multiplicity ◽  
Tsallis Statistics ◽  
Gamma Distributions ◽  
Scaling Hypothesis ◽  
Target Experiment ◽  
Multiplicity Distributions ◽  
Energy Hadron ◽  
Experimental Values

The study of higher-order moments of a distribution and its cumulants constitute a sensitive tool to investigate the correlations between the particle produced in high-energy interactions. In our previous work, we have used the Tsallis [Formula: see text] statistics, NBD, Gamma and shifted Gamma distributions to describe the multiplicity distributions in [Formula: see text]-nucleus and [Formula: see text]-nucleus fixed target interactions at various energies ranging from [Formula: see text][Formula: see text]GeV to 800[Formula: see text]GeV. In this study, we have extended our analysis by calculating the moments using the Tsallis model at these fixed target experiment data. By using the Tsallis model, we have also calculated the average charged multiplicity and its dependence on energy. It is found that the average charged multiplicity and moments predicted by the Tsallis statistics are in much agreement with the experimental values and indicates the success of the Tsallis model on data from visual detectors. The study of moments also illustrates that KNO scaling hypothesis holds good at these energies.

A UNIFIED VIEW OF MULTIPLICITY DISTRIBUTIONS BASED ON A PURE BIRTH PROCESS

1994 ◽  
Vol 09 (36) ◽  
pp. 3359-3366 ◽  
Author(s):  
S. CHATURVEDI ◽  
V. GUPTA ◽  
S.K. SONI
Keyword(s):  
Negative Binomial ◽  
Initial Conditions ◽  
Probability Distributions ◽  
High Energy ◽  
Particle Multiplicity ◽  
Charged Particle Multiplicity ◽  
Birth Process ◽  
Unified View ◽  
High Energy Collisions ◽  
Multiplicity Distributions

Various probability distributions which have been proposed to explain the charged particle multiplicity distributions in high energy collisions are shown to arise from the evolution equation of a pure birth process subject to appropriate initial conditions. For example, both the negative binomial distribution (NBD) as well as the partially coherent laser distribution (PCLD) can be obtained in this way. New interrelations between some of these probability distributions are also brought out.

scholarly journals On the Sums of Compound Negative Binomial and Gamma Random Variables

2009 ◽  
Vol 46 (1) ◽  
pp. 272-283 ◽  
Author(s):  
P. Vellaisamy ◽  
N. S. Upadhye
Keyword(s):  
Graph Theory ◽  
Shortest Path ◽  
Negative Binomial ◽  
Random Variables ◽  
Random Parameter ◽  
Exact Expression ◽  
Shortest Path Problem ◽  
Binomial Distributions ◽  
Parameter Representation ◽  
Gamma Distributions

We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.

A generalised incomplete beta function and its application to multi-line stock control

1973 ◽  
Vol 10 (04) ◽  
pp. 748-760 ◽  
Author(s):  
J. C. Gittins ◽  
M. J. Maher
Keyword(s):  
Distribution Function ◽  
Negative Binomial ◽  
Beta Function ◽  
Distribution Functions ◽  
Incomplete Beta Function ◽  
Stock Control ◽  
Binomial Distributions ◽  
The Family ◽  
Optimal Values ◽  
Poisson Demands

The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.

NEGATIVE BINOMIAL FITS TO MULTIPLICITY DISTRIBUTIONS FROM CENTRAL 16O+Cu COLLISIONS AT 14.6 A GeV/c AND THEIR IMPLICATION FOR “INTERMITTENCY”

1994 ◽  
Vol 09 (02) ◽  
pp. 89-100 ◽  
Author(s):  
M.J. TANNENBAUM
Keyword(s):  
Transverse Energy ◽  
Negative Binomial ◽  
Heavy Ion ◽  
Linear Increase ◽  
Particle Multiplicity ◽  
Charged Particle Multiplicity ◽  
Factorial Moments ◽  
Ion Collisions ◽  
Binomial Distributions ◽  
Multiplicity Distributions

The method of normalized factorial moments has been used extensively to study the fluctuations in pseudorapidity of charged particle multiplicity as a function of the interval δη. Experience in analyzing the data from light and heavy ion collisions in terms of distributions rather than moments suggests that conventional fluctuations of multiplicity and transverse energy can be well described by gamma or Negative Binomial Distributions (NBD). Multiplicity distributions from central (ZCAL) collisions of 16 O+Cu at 14.6 A GeV/c have been analyzed by the E802 collaboration as a function of the interval δη≥0.1 in the range 1.2≤η≤2.2. Excellent fits to NBD were obtained in all δη bins. The k parameter of the NBD fit exhibits a steep linear increase with the δη interval, which due to the well known property of the NBD under convolution, indicates that the multiplicity distributions in adjacent bins of pseudorapidity δη~0.1 are largely statistically independent. This result explains and demystifies “intermittency.”

RAPIDITY DEPENDENCE OF MULTIPLICITY DISTRIBUTIONS IN ALPHA-EMULSION INTERACTIONS AT 12.4A GEV

1990 ◽  
Vol 05 (20) ◽  
pp. 3985-3997 ◽  
Author(s):  
M. M. AGGARWAL ◽  
R. ARORA ◽  
S. B. BERI ◽  
V. S. BHATIA ◽  
M. KAUR ◽  
...  
Keyword(s):  
Size Dependence ◽  
Negative Binomial ◽  
Target Size ◽  
Binomial Distributions ◽  
Multiplicity Distributions ◽  
Rapidity Dependence ◽  
Forward Hemisphere

We present results based on an analysis of alpha-emulsion interactions at 12.4A GeV. The multiplicity distributions of shower particles in the restricted rapidity intervals are well described by negative binomial distributions (NBD). The behaviour of parameters [Formula: see text] and 1/k of the NBD with increasing Δy in the backward hemisphere is quite different from that found in elementary collisions. Star size dependence is also investigated. An attempt has been made to interpret the results in the framework of a clan model. In the forward hemisphere the average decay multiplicity of clans is small [Formula: see text] and seems to be target-independent, whereas bigger clans are produced in the backward hemisphere and their sizes seem to depend on the target size. Also the average number of clans in the backward hemisphere is less than that in the forward hemisphere and shows target size dependence. Clans seem to be produced independently and their extent in rapidity seems to be target-independent.

scholarly journals On the Sums of Compound Negative Binomial and Gamma Random Variables

2009 ◽  
Vol 46 (01) ◽  
pp. 272-283 ◽  
Author(s):  
P. Vellaisamy ◽  
N. S. Upadhye
Keyword(s):  
Graph Theory ◽  
Shortest Path ◽  
Negative Binomial ◽  
Random Variables ◽  
Random Parameter ◽  
Exact Expression ◽  
Shortest Path Problem ◽  
Binomial Distributions ◽  
Parameter Representation ◽  
Gamma Distributions

We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability ofm-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.

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