Topic
Probability-generating function
About: Probability-generating function is a research topic. Over the lifetime, 752 publications have been published within this topic receiving 9361 citations.
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TL;DR: The main result gives the time-dependent form of the first and second factorial moments of the counting process, which is represented by eigenvalues and eigenvectors of the matrix generating function of the batch size.
Abstract: Consider a batch Markovian arrival process (BMAP) as the counting process of an underlying Markov process representing the state of environment. Such a process is useful for representing correlated inputs for example. They are used both as a modeling tool and as a theoretical device to represent and approximate superposition of input processes and complex large systems. Our objective is to consider the first and second moments of the counting process depending on time and state. Assuming that the probability generating functions of batch size are analytic, and that eigenvalues of the infinitesimal generator are simple, we derive an analytic diagonalization for the matrix generating function of the counting process. Our main result gives the time-dependent form of the first and second factorial moments of the counting process, which is represented by eigenvalues and eigenvectors of the matrix generating function of the batch size.
14 citations
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TL;DR: In this paper, Aly and Benkherouf presented a new family of distributions based on probability generating functions and derived a very useful representation for the Harris extended density function as an absolutely convergent power series of the survival function of the baseline distribution.
Abstract: A new method for generating new classes of distributions based on the probability-generating function is presented in Aly and Benkherouf [A new family of distributions based on probability generating functions. Sankhya B. 2011;73:70–80]. In particular, they focused their interest to the so-called Harris extended family of distributions. In this paper, we provide several general results regarding the Harris extended models such as the general behaviour of the failure rate function. We also derive a very useful representation for the Harris extended density function as an absolutely convergent power series of the survival function of the baseline distribution. Additionally, some stochastic order relations are established and limiting distributions of sample extremes are also considered for this model. These general results are illustrated in several special Harris extended models. Finally, we discuss estimation of the model parameters by the method of maximum likelihood and provide an application to real da...
14 citations
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TL;DR: It is shown that first two waiting time probability density functions can reproduce the results of the ordinary and fractional diffusion equations for all the time regions from small to large times but the third one shows a much more complicated pattern.
Abstract: We derive an integrodifferential diffusion equation for decoupled continuous time random walk that is valid for a generic waiting time probability density function and external force. Using this equation we also study diffusion behaviors for a couple of specific waiting time probability density functions such as exponential, a combination of power law and generalized Mittag-Leffler function and a sum of exponentials under the influence of a harmonic trap. We show that first two waiting time probability density functions can reproduce the results of the ordinary and fractional diffusion equations for all the time regions from small to large times. But the third one shows a much more complicated pattern. Furthermore, from the integrodifferential diffusion equation we show that the second Einstein relation can hold for any waiting time probability density function.
14 citations