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.

Automated design of probability distributions as mutation operators for evolutionary programming using genetic programming

Abstract: The mutation operator is the only source of variation in Evolutionary Programming. In the past these have been human nominated and included the Gaussian, Cauchy, and the Levy distributions. We automatically design mutation operators (probability distributions) using Genetic Programming. This is done by using a standard Gaussian random number generator as the terminal set and and basic arithmetic operators as the function set. In other words, an arbitrary random number generator is a function of a randomly (Gaussian) generated number passed through an arbitrary function generated by Genetic Programming. Rather than engaging in the futile attempt to develop mutation operators for arbitrary benchmark functions (which is a consequence of the No Free Lunch theorems), we consider tailoring mutation operators for particular function classes. We draw functions from a function class (a probability distribution over a set of functions). The mutation probability distribution is trained on a set of function instances drawn from a given function class. It is then tested on a separate independent test set of function instances to confirm that the evolved probability distribution has indeed generalized to the function class. Initial results are highly encouraging: on each of the ten function classes the probability distributions generated using Genetic Programming outperform both the Gaussian and Cauchy distributions.

Estimation of Parameters on Some Extensions of the Katz Family of Discrete Distributions Involving Hypergeometric Functions

Abstract: A two-parameter family of discrete distributions developed by Katz (1963) is extended to three- and four-parameter families whose probability generating functions involve hypergeometric functions. This extension contains other distributions appearing in the literature as particular cases. Various methods of estimating the parameters are investigated and their asymptotic efficiency relative to maximum likelihood estimators compared.

A first passage time distribution for a discrete version of the Ornstein–Uhlenbeck process

Abstract: The probability of the first entrance to the negative semi-axis for a one-dimensional discrete Ornstein–Uhlenbeck (O-U) process is studied in this work. The discrete O-U process is a simple generalization of the random walk and many of its statistics may be calculated using essentially the same formalism. In particular, the case in which Sparre-Andersen's theorem applies for normal random walks is considered, and it is shown that the universal features of the first passage probability do not extend to the discrete O-U process. Finally, an explicit expression for the generating function of the probability of first entrance to the negative real axis at step n is calculated and analysed for a particular choice of the step distribution.

The Analysis and Optimization of Probability Functions

Abstract: A problem of probability function optimization is considered. This function represents probability that some random quantity depending on deterministic parameters does not exceed some given level. The problem is motivated by studies of safety domains and risk control problems in complex stochastic systems. For example, pollution control includes maximization of probability that some given levels of deposition at reception points are not exceeded. Optimization of probability function is performed over a given range of parameters. To solve the problem stochastic quasi-gradient method is applied under quasi-concavity assumption on functions and measures involved. Convergence and rate of convergence results are presented.