Lagrangian Probability Distributions
Lagrangian Probability Distributions
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DOI：
10.1007/0-8176-4477-6_2
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发表时间：
2005-12
期刊：
影响因子：
--
通讯作者：
P. Consul;F. Famoye
中科院分区：
文献类型：
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作者：
P. Consul;F. Famoye
离散的概率分布拟合到观察到的计数，以找到一个模式，这可能会导致一个调查员，看看是否可以建立一些生成模型的过程中的研究。由于每一种现象都是多变量的，因此找到正确模型的任务变得困难。几乎所有的生物、心理、社会、农业或自然过程的模型都是近似的。因此，每个模型都是真实的生活的简化。可能，每个观察到的模式都是某个随机过程的稳定状态。由于世界上不同的研究人员在各自的领域面临着观察计数，他们试图在他们的观察中找到一个特定的模式，他们的努力导致他们发现新的和更复杂的概率分布。现在有大量的离散概率分布，它们被分为各种类别、族和广义单变量、双变量和多变量离散分布。关于这些离散概率分布及其重要性质的详细说明可以在Balakrishnan和Nevzorov（2003）、约翰逊、Kotz和Balakrishnan（1997）、约翰逊、Kotz和肯普（1992）、Patil等人的著名著作中找到。（1984）、Patil和Joshi（1968）、约翰逊和Kotz（1969）、Mardia（1970）、Ord（1972）以及其他许多人。当这本书准备付印时，我们被告知这本由约翰逊、科茨和肯普（1992）合著的书正在修订中。离散拉格朗日概率分布形成了一个非常大的和重要的类，其中包含许多家庭的概率分布。这些概率分布非常有用，因为它们在各个领域中经常出现。大多数关于这些分布的研究工作只能在期刊上找到。广义Poisson分布是Consul（1989 a）详细讨论过的一个重要模型。约翰逊、科茨和肯普（1992）在他们的书的五章中以不同的标题描述了拉格朗日概率分布类的一些族。拉格朗日分布类中最重要的是拉格朗日变换z= ug（z），由拉格朗日（1736-1813）给出，他用它来表示z为u的幂级数，然后将函数f（z）展开为u的幂级数，如第1章所述。Otter（1949）是第一个认识到这种转化对乘法过程发展的重要性的人。如果g（z）是从具有多个顶点的根树中的任何顶点开始的段的数量的pgf，则变换z= ug（z）提供根树中的顶点的数量的pgf。Otter（1949）指出，树中n段之后的顶点数可以解释为分支过程的第n代中的成员数，并且它可以用于流行病的研究，
Discrete probability distributions are fitted to observed counts to find a pattern which may lead an investigator to see if some generating models can be set up for the process under study. As every phenomenon is of a multivariate nature, the task of finding the correct model becomes difficult. Practically all models for biological, psychological, social, agricultural, or natural processes are approximations. Thus, every model is a simplification of real life. Possibly, every observed pattern is the steady state of some stochastic process. As different researchers in the world are faced with observed counts in their respective fields and they try to find a specific pattern in their observations, their efforts lead them to the discovery of new and more complex probability distributions. A large number of discrete probability distributions are now available which are divided into various classes, families, and generalized univariate, bivariate, and multivariate discrete distributions. An extensive account of these discrete probability distributions and their important properties can be found in the well-known works by Balakrishnan and Nevzorov (2003), Johnson, Kotz, and Balakrishnan (1997), Johnson, Kotz, and Kemp (1992), Patil, et al.(1984), Patil and Joshi (1968), Johnson and Kotz (1969), Mardia (1970), Ord (1972), and many others. As this book was ready to go to press, we were informed that the book by Johnson, Kotz, and Kemp (1992) was under revision. Discrete Lagrangian probability distributions form a very large and important class which contains numerous families of probability distributions. These probability distributions are very useful because they occur quite frequently in various fields. Most of the research work on these distributions is available in journals only. The generalized Poisson distribution is one important model which has been discussed in detail by Consul (1989a). Johnson, Kotz, and Kemp (1992) have described some families of the class of Lagrangian probability distributions under different titles in five chapters of their book. The prime source of importance in the class of Lagrangian distributions is the Lagrange transformation z= ug (z), given by Lagrange (1736–1813) and used by him to express z as a power series in u and then to expand a function f (z) into a power series of u as described in Chapter 1. Otter (1949) was the first person who realized the importance of this transformation for the development of a multiplicative process. If g (z) is the pgf of the number of segments from any vertex in a rooted tree with a number of vertices, then the transformation z= ug (z) provides the pgf for the number of vertices in the rooted tree. Otter (1949) showed that the number of vertices after n segments in a tree can be interpreted as the number of members in the nth generation of a branching process and that it can be used in the study of epidemics, spread of