负超几何随机变量期望与方差的一种简捷计算

给出了负超几何分布的概率模型,通过将负超几何分布随机变量进行和式分解,比较简捷地计算了它的期望和方差,并指出文献[4]计算的期望和方差是错误的.     

离散型随机变量高阶原点矩的一种新的递推计算方法

针对服从二项、泊松、几何、负二项、超几何、负超几何以及对数级数分布等离散型随机变量,给出了求其高阶原点矩的一个较为简单的递推计算方法.不仅非常容易地求出这些离散型随机变量的高阶原点矩,避免了计算阶乘矩或求导等复杂的运算,而且便于学生理解.论文还给出了这些离散型随机变量的3阶和4阶原点矩的表达式.     

超几何分布的数字特征和概率计算的探讨

用定义法、性质法、概率母函数法三种方法探索了超几何分布的数学期望和方差的求法,同时又给出了超几何分布、二项分布、泊松分布和正态分布之间的近似关系,从而解决了超几何分布的概率计算问题.     

几何分布和负二项分布高阶矩的递推公式

给出几何分布、负二项分布高阶矩的递推公式.     

超几何分布的数学期望和方差的定义求法

利用排列的性质,从超几何分布的定义出发,针对服从超几何分布的随机变量,给出直接计算其数学期望和方差的一种方法．     

超几何分布精确值的计算方法

一、问题的提出 1.超几何分布的模型 设有一批产品,批量为N,其中不合格品M件.如果从该批中随机地一次抽取一个容量为n的样本,则样本中不合格品的件数(记为X)是一个随机变量且服从超几何分布,记其一次观察值为x,那么分布律为其中为非负整数,且 表示从Z个不同元素中取出y个元素的     

离散寿命分布类的统计封闭性

由于几何分布的无记忆性,使得几何分布在离散型寿命分布研究中起着极其重要的作用.文章给出几何分布的一个重要特性———其第一个次序统计量仍服从几何分布(称其满足统计封闭特性).针对不同的离散型寿命分布类研究其是否具有统计封闭性.     

边缘分布条件分布的几何意义及推广

从几何的角度,揭示边缘分布条件分布的几何意义,浅显易懂,帮助学生深入正确理解定义所隐含的内容,同时,进一步推广了边缘分布和条件分布.     

伯努利数和二项分布及超几何分布的快速计算

本文利用伯努利数建立了二项分布值和超几何分布值的快速计算公式，这些公式计算的结果精确度高，而且非常便于计算机编程．     

对数伽玛分布与负对数伽玛分布的条件概率封闭性

设X服从以α和λ为参数的对数伽玛分布或负对数伽玛分布,V服从以β为参数的负幂分布,则在X相似文献   

A Weighted Likelihood Ratio of Two Related Negative Hypergeomeric Distributions

In this paper we consider some related negative hypergeometric distributions arising from the problem of sampling without replacement from an urn containing balls of different colours and in different proportions but stopping only after some specific number of balls of different colours have been obtained. With the aid of some simple recurrence relations and identities we obtain in the case of two colours the moments for the maximum negative hypergeometric distribution, the minimum negative hypergeometric distribution,the likelihood ratio negative hypergeometric distribution and consequently the likelihood proportional negative hypergeometric distributiuon. To the extent that the sampling scheme is applicable to modelling data as illustrated with a biological example and in fact many situations of estimating Bernoulli parameters for binary traits within a finite population, these are important first-step results.     

Risk aggregation in multivariate dependent Pareto distributions

In this paper we obtain closed expressions for the probability distribution function of aggregated risks with multivariate dependent Pareto distributions. We work with the dependent multivariate Pareto type II proposed by Arnold (1983, 2015), which is widely used in insurance and risk analysis. We begin with an individual risk model, where the probability density function corresponds to a second kind beta distribution, obtaining the VaR, TVaR and several other tail risk measures. Then, we consider a collective risk model based on dependence, where several general properties are studied. We study in detail some relevant collective models with Poisson, negative binomial and logarithmic distributions as primary distributions. In the collective Pareto–Poisson model, the probability density function is a function of the Kummer confluent hypergeometric function, and the density of the Pareto–negative binomial is a function of the Gauss hypergeometric function. Using data based on one-year vehicle insurance policies taken out in 2004–2005 (Jong and Heller, 2008) we conclude that our collective dependent models outperform other collective models considered in the actuarial literature in terms of AIC and CAIC statistics.     

An inequality for multiple convolutions with respect to Dirichlet probability measure

A sharp multiple convolution inequality with respect to Dirichlet probability measure on the standard simplex is presented. Its discrete version in terms of the negative binomial coefficients is proved as well. The new bounds for the Dirichlet distribution and iterated convolutions are obtained as the consequences of the main result. Also some binomial, exponential, and generalized hypergeometric applications are discussed.     

On characterization of power series distributions by a marginal distribution and a regression function

The conditional distribution of Y given X=x, where X and Y are non-negative integer-valued random variables, is characterized in terms of the regression function of X on Y and the marginal distribution of X which is assumed to be of a power series form. Characterizations are given for a binomial conditional distribution when X follows a Poisson, binomial or negative binomial, for a hypergeometric conditional distribution when X is binomial and for a negative hypergeometric conditional distribution when X follows a negative binomial.     

Approximation of inverse moments of discrete distributions

For a wide class of discrete distributions, we derive a representation of the inverse (negative) moments through the Stirling numbers of the first kind and inverse factorial moments. We specialize the results for the Poisson, binomial, hypergeometric and negative binomial distributions.     

Certain estimation problems for multivariate hypergeometric models

Certain estimation problems associated with the multivariate hypergeometric models: the property of completeness, maximum likelihood estimates of the parameters of multivariate negative hypergeometric, multivariate negative inverse hypergeometric, Bayesian estimation of the parameters of multivariate hypergeometric and multivariate inverse hypergeometrics are discussed in this paper. A two stage approach for generating the prior distribution, first by setting up a parametric super population and then choosing a prior distribution is followed. Posterior expectations and variances of certain functions of the parameters of the finite population are provided in cases of direct and inverse sampling procedures. It is shown that under extreme diffuseness of prior knowledge the posterior distribution of the finite population mean has an approximate mean and variance (N-n)S ²/Nn, providing a Bayesian interpretation for the classical unbiased estimates in traditional sample survey theory.     

On the logistic midrange

Summary It is well-known that for a large family of distributions, the sample midrange is asymptotically logistic. In this article, the logistic midrange is closely examined. Its distribution function is derived using Dixon's formula (Bailey (1935,Generalized Hypergeometric Series, Cambridge University Press, p. 13)) for the generalized hypergeometric function with unit argument, together with appropriate techniques for the inversion of (bilateral) Laplace transforms. Several relationships in distribution are established between the midrange and sample median of the logistic and Laplace random variables. Possible applications in testing for outliers are also discussed.     

Simple and efficient algorithms for computing exact cumulative discrete probabilities and its inverses

Summary  This paper deals with the computation of exact cumulative probabilities of discrete distributions and its inverses. For the computation of cumulative probabilities an efficient and universal algorithm of 15 lines is presented, which can be applied to the most important discrete distributions (e.g. the binomial, the poisson and the hypergeometric distribution). With a slight modification an algorithm of 20 lines is obtained for the calculation of the respective inverse distributions. The accuracy of both algorithms can be specified. Both algorithms are simple, very fast and numerically stable even if the sample size is one billion.