自主探究水到渠成——二项分布模型构建的数学设计

二项分布是继超几何分布后的又一应用广泛的概率模型,而超几何分布在产品数量n相当大时可以近似地看成二项分布.在自然现象和社会现象中,大量的随机变量都服从或近似地服从二项分布,实际应用广泛,理论上也非常重要.     

通过随机变量刻画随机现象加深理解随机思想(续)

4.4二项分布与超几何分布 二项分布是最常见的一种离散型分布,19世纪以前的概率统计可以说就是二项分布的天下.我们知道,保险业是最早应用概率论的领域,在有关保险的问题中涉及大量的二项分布计算问题.另外,在很长时期内,统计方法在社会问题中的应用主要限于人口统计,特别是出生的男、女婴儿的性别比例问题,这是一个典型的二项分布...     

从"和值14"现象谈二项分布的典型应用

分析了"和值14"现象产生的主要原因是不少彩民应用二项分布知识于彩票,实际上概率在数字彩中的作用并不太明显.但二项分布在其他领域中却有着广泛的应用,譬如在人力资源管理和风险管理上就有典型的应用.     

超几何分布和二项分布的区别

<正>超几何分布和二项分布两个应用广泛的概率模型是高中数学概率章节的重要内容,学生在解题过程中往往因为不能正确辨析两者之间的区别而张冠李戴,所以帮助学生充分理解超几何分布和二项分布的概率模型及特点和应用条件,才能对超几何分布和二项分布有较深刻的理性认识,才能够解释、举例并能准确解决有关问题.下面笔者就超几何分布和二     

二项分布的参数估计问题研究

本文主要讨论二项分布的参数估计问题.对GB/T 4087.1-1983,GB/T 4087.2-1983,GB/T4087.3-1985给出的经典二项分布参数点估计、区间估计和可靠度置信下限计算方法进行了分析,指出了其中存在的问题.根据二项分布的数学表达式推导出了二项分布参数的概率分布密度函数,在此基础上提出了进行二项分布参数估计的一般方法.     

以生活实际为背景的二项分布

<正>二项分布是高中人教A版选修2-3的知识点,是一个应用广泛的离散型随机变量的概率模型.下面以几个具体实例来阐述二项分布的应用,将有助于我们更好的理解二项分布的实际意义.问题1 (2010,湖南卷,理17)如图1是某城市通过抽样得到的居民某年月均用水量(单位:吨)的频率分布直方图.若将频率视为概率,从这个城市随机抽取3位居民(看作有放回抽样),求月均用水量在3至4吨的居民数X的分布列和数学期望.     

索赔次数数据分布的拟合方法的分析和比较

泊松分布和负二项分布常用于拟合保险索赔次数.它们和二项分布统称为(a,b,0)分布族.本文对(a,b,0)分布族进行了研究,然后在此基础上给出了(a,b,0)分布离散型随机变量是服从泊松分布,还是服从负二项分布或二项分布的检验方法.本文基于我国某家保险公司的索赔次数数据进行了实证分析,并对检验的功效进行了模拟研究.     

负二项分布随机变量的分解定理

给出了负二项分布的分解定理,并进一步研究了负二项分布的有关性质和近似计算公式.在复杂排队系统中离散状态下顾客等待时间分布的概率计算中起到了重要作用.     

考虑个体保单风险特征的最优奖惩系统

奖惩系统在汽车保险中的应用非常普遍。论文首先介绍和讨论了泊松-伽马假设下的最优奖惩系统及其性质;其次在假设个体保单的索赔频率服从二项分布,而二项分布的一个参数服从贝塔分布的条件下,建立了一种考虑个体保单风险特征信息的最优奖惩系统,其中风险特征信息可以通过广义线性模型的形式引入奖惩系统;然后在假设个体保单的索赔频率服从负二项分布,而负二项分布的一个参数服从贝塔分布的条件下,建立了另一个最优奖惩系统;最后讨论了这两个奖惩系统的性质和应用。     

探究服从二项分布的随机变量概率最大值问题

二项分布是应用最广泛的离散型随机变量概率模型．对与二项分布有关的一些问题的探究是很有意义的．同学们会提出这样的问题：服从二项分布的随机变量取何值时概率最大？下面我们来探究一下．     

“f二项近似式”在蚕种检验计算中的应用

本文对蚕种检验中的超几何分布用“ｆ二项近似式”进行计算,既方便,又准确率高。     

彩票模式设计的最优化研究

讨论了彩票中一种常用模式设计的最优化问题 .特别是为描述某一彩票模式的吸引力 ,我们根据各奖项出现的概率、奖金额等因素引进了效用函数 U,对彩票模式的合理性进行评价 ;建立了以彩票管理部门的收益为目标函数、合理性为约束的优化模型 ,并给出了两个对彩票管理部门更为有利、合理的彩票模式 .     

负二项分布两种参数估计及其比较

负二项分布在昆虫空间分布中有重要应用,其参数常用矩法或零频率法来估计。本文讨论两种估计的性质,得到大样本情形下,零频率较大时,零频率估计较矩估计有更高精度的结论;通过计算机模拟,验证结论。     

Estimation of parameters in the beta binomial model

This paper contains some alternative methods for estimating the parameters in the beta binomial and truncated beta binomial models. These methods are compared with maximum likelihood on the basis of Asymptotic Relative Efficiency (ARE). For the beta binomial distribution a simple estimator based on moments or ratios of factorial moments has high ARE for most of the parameter space and it is an attractive and viable alternative to computing the maximum likelihood estimator. It is also simpler to compute than an estimator based on the mean and zeros, proposed by Chatfield and Goodhart (1970,Appl. Statist.,19, 240–250), and has much higher ARE for most part of the parameter space. For the truncated beta binomial, the simple estimator based on two moment relations does not behave quite as well as for the BB distribution, but a simple estimator based on two linear relations involving the first three moments and the frequency of ones has extremely high ARE. Some examples are provided to illustrate the procedure for the two models.     

Statistical Inference for the Rayleigh distribution under progressively Type-II censoring with binomial removal

This paper takes into account the estimation for the unknown parameter of the Rayleigh distribution under Type II progressive censoring with binomial removals, where the number of units removed at each failure time follows a binomial distribution. Maximum likelihood and Bayes procedure are used to derive both point and interval estimates of the parameters involved in the model. The expected termination point to complete the censoring test is computed and analyzed under binomial censoring scheme. Numerical examples are given to illustrate the approach by means of Monte Carlo simulation. A real life data set is used for illustrative purposes in conclusion.     

Bernshtein's regularity conditions in a problem of empirical bayesian approach

It is proved that the posterior distribution of the random success parameter in a binomial distribution may be approximated by a beta distribution, if the second parameter of the binomial lawn, goes to infinity and the prior density of the success parameter belongs to some subset of, 1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 79, pp. 38–43, 1978.In conclusion, I would like to thank L. N. Bol'shev for suggesting the problem in this form in one of the lectures at the Steklov Mathematical Institute.     

Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k

A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer less than the specified length of the runs, an enumeration scheme called -overlapping way of counting is defined. Let k and be positive integers satisfying < k. Based on independent Bernoulli trials, it is shown that the number of (– 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length follows the generalized binomial distribution of order (k–). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k– 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.     

Two-Parametric Compound Binomial Approximations

We consider two-parametric compound binomial approximation of the generalized Poisson binomial distribution. We show that the accuracy of approximation essentially depends on the symmetry or shifting of distributions and construct asymptotic expansions. For the proofs, we combine the properties of norms with the results for convolutions of symmetric and shifted distributions. In the lattice case, we use the characteristic function method. In the case of almost binomial approximation, we apply Steins method.__________Published in Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 443–466, October–December, 2004.     

The beta stochastic utility (β-SU)

In this article, a new model for decision making under uncertainty is presented. Here, we model human attitude toward risks to show that an individual estimate of the expected utility of a lottery follows a generalized Beta distribution with a random error that follows a similar distribution. An individual is said to maximize his stochastic utility when requested to present his preference between risky lotteries. Hypothetically, risky lotteries are those exhibiting wider ranges of rewards where human estimate will not be below the utility of the lowest reward nor above the highest of the lottery. The Beta distribution is bounded and complies to such intuitive preconditions with a variance depending on such bounds. The proposed model will overestimate/underestimate the expected utility of a lottery according to the lottery probability mass and individuals' risk attitudes. By such estimation, our model conforms to the fourfold choice pattern. The model also explains the violations present as inconsistencies in the expected utility theory, such as Allais paradox, common consequence effect, common ratio effect, and the violation of betweenness that can be found in the fourfold choice pattern. For the validation purposes, 13 datasets from literature were collected and tested. The β-SU model fits the data at least as good as other approaches such as the CPT/StEUT and presents higher prediction log-likelihoods and less sum of squared errors in most of the cases, a matter that supports the proposition that human estimates of the expected utility may be drawn out of a generalized Beta distribution.