Asymptotics of the Hurwitz Binomial Distribution Related to Mixed Poisson Galton–Watson Trees

2001 ◽  
Vol 10 (3) ◽  
pp. 203-211 ◽  
Author(s):  
JÜRGEN BENNIES ◽  
JIM PITMAN
Keyword(s):  
Probability Distribution ◽  
Asymptotic Behaviour ◽  
Binomial Distribution ◽  
Random Tree ◽  
Binomial Theorem ◽  
Offspring Distribution

Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of non-root vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this distribution is described in a limiting regime in which the fringe subtree converges in distribution to a Galton–Watson tree with a mixed Poisson offspring distribution.

On population-size-dependent branching processes

1984 ◽  
Vol 16 (1) ◽  
pp. 30-55 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Stochastic Model ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Population Size ◽  
Gamma Distribution ◽  
Branching Processes ◽  
Independent Random Variables ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

A Simple Modification of the Binomial Distribution

1960 ◽  
Vol 15 (06) ◽  
pp. 436-444
Author(s):  
S. W. Dharmadhikari
Keyword(s):  
Probability Distribution ◽  
Poisson Distribution ◽  
Binomial Distribution ◽  
Probability Distributions ◽  
Simple Modification ◽  
Type Iii

Given any probability distribution, new distributions can be derived from it by assuming its parameters to follow some specific probability distributions. A simple example of this process is provided by the Poisson distributionP(r∣λ) =e-λλr/r! (r= o, 1, 2, …).If the parameterλis assumed to follow the Pearson's Type III lawthen the probability ofrsuccesses is obtained as

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.

scholarly journals A mechanistic model for the negative binomial distribution of single-cell mRNA counts

2019 ◽  
Author(s):  
Lisa Amrhein ◽  
Kumar Harsha ◽  
Christiane Fuchs
Keyword(s):  
Steady State ◽  
Probability Distribution ◽  
Single Cell ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Probability Distributions ◽  
Mechanistic Model ◽  
Steady State Probability ◽  
Degradation Models

SummarySeveral tools analyze the outcome of single-cell RNA-seq experiments, and they often assume a probability distribution for the observed sequencing counts. It is an open question of which is the most appropriate discrete distribution, not only in terms of model estimation, but also regarding interpretability, complexity and biological plausibility of inherent assumptions. To address the question of interpretability, we investigate mechanistic transcription and degradation models underlying commonly used discrete probability distributions. Known bottom-up approaches infer steady-state probability distributions such as Poisson or Poisson-beta distributions from different underlying transcription-degradation models. By turning this procedure upside down, we show how to infer a corresponding biological model from a given probability distribution, here the negative binomial distribution. Realistic mechanistic models underlying this distributional assumption are unknown so far. Our results indicate that the negative binomial distribution arises as steady-state distribution from a mechanistic model that produces mRNA molecules in bursts. We empirically show that it provides a convenient trade-off between computational complexity and biological simplicity.Graphical Abstract

Probability and distributions

2020 ◽  
pp. 243-280
Author(s):  
Janet L. Peacock ◽  
Philip J. Peacock
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Poisson Distribution ◽  
Binomial Distribution ◽  
Probability Distributions ◽  
Medical Statistics

Probability and probability distributions play a central part in medical statistics. This chapter defines what is meant by probability and describes the rules by which probabilities are combined. It then describes how the use of probability leads to the concept of a probability distribution and shows how these distributions are used in medical statistics. Examples are given of the use of key distributions: the Normal distribution, the binomial distribution, and the Poisson distribution.

scholarly journals Convergence properties of the q-deformed binomial distribution

2010 ◽  
Vol 4 (1) ◽  
pp. 66-80 ◽  
Author(s):  
Martin Zeiner
Keyword(s):  
Asymptotic Behaviour ◽  
Exponential Distribution ◽  
Binomial Distribution ◽  
Convergence Properties ◽  
Convergence Results

We consider the q-deformed binomial distribution introduced by S.C. Jing 1994: The q-deformed binomial distribution and its asymptotic behaviour, J. Phys. A 27 (2) (1994), 493{499 and W.S. Chung et al.: q-deformed probability and binomial distribution, Internat. J. Theoret. Phys. 34 (11) (1995), 2165{2170 and establish several convergence results involving the Euler and the exponential distribution; some of them are q-analogues of classical results.

On population-size-dependent branching processes

1984 ◽  
Vol 16 (01) ◽  
pp. 30-55 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Stochastic Model ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Population Size ◽  
Gamma Distribution ◽  
Branching Processes ◽  
Independent Random Variables ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a stochastic model for the development in time of a population {Z n } where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m &lt; l, m = 1, m&gt; l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m &lt; 1 or m = 1 and mn approaches 1 not slower than n –2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n –1, then Zn /n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an , such that Zn /an converges in probability to a non-degenerate limit. If mn approaches m &gt; 1 not slower than n– α, α &gt; 0, and do not grow to ∞ faster than nß , β &lt;1 then Zn /mn converges almost surely and in L 2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

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