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.

Locally sub-gaussian random variables and the strong law of large numbers

Abstract: In this paper we generalize the concept of sub-Gaussian random variable to that of “locally” sub-Gaussian random variable. Some properties of locally sub-Gaussian random variables are presented. It is shown that a “local” version of the moment inequality used by Taylor and Hu in 1987 can be used to give an equally simple proof of the strong law of large numbers for locally sub-Gaussian random variables.

Bias calculations for adaptive urn designs

Abstract: A clinical trial model is considered in which two treatments with immediate binary responses are to be compared. An adaptive urn design is used to assign patients to the treatments. The bias and variance of the maximum likelihood estimators of the probabilities of success are derived by differentiating the fundamental identity of sequential analysis. By embedding the design in a continuous-time process, probability generating functions are then calculated to obtain approximations for the bias and variance. Simulation is used to assess the accuracy of the approximations. It is shown that the bias cannot be ignored, and that the adaptive rules which are subcritical in nature have the most mathematically tractable bias and are the least variable. Methods for correcting for the bias are also addressed.

The probabilistic solutions to nonlinear random vibrations of multi-degree-of-freedom systems

Abstract: The probability density function of the responses of nonlinear random vibration of a multi-degree-of-freedom system is formulated in the defined domain as an exponential function of polynomials in state variables. The probability density function is assumed to be governed by Fokker-Planck-Kolmogorov (FPK) equation. Special measure is taken to satisfy the FPK equation in the average sense of integration with the assumed function and quadratic algebraic equations are obtained for determining the unknown probability density function. Two-degree-of-freedom systems are analyzed with the proposed method to validate the method for nonlinear multi-degree-of-freedom systems. The probability density functions obtained with the proposed method are compared with the obtainable exact and simulated ones. Numerical results showed that the probability density function solutions obtained with the presented method are much closer to the exact and simulated solutions even for highly nonlinear systems with both external and parametric excitations.

A lofistic rlgression analysis of response-time data where the hazard function is time dependent

Abstract: A logistic-exponential model for analyzing response-time data involving regressor variables is modified to allow for non-consrarey of the hazard function. For the discrete observation case illustrated the logit of the probability of responding in a time interval cf arbitrary length is taken as the sum of a function of resressor variables and a function of the time variable. The particular functions chosen in the two medical examples analyzed are linear in the parameters involved. A polynomial function of time is employed in the absence of knowledge as to a more appropriate form. Various issues arising in the analysis made are discussed.