Topic

# Probability-generating function

About: Probability-generating function is a(n) research topic. Over the lifetime, 752 publication(s) have been published within this topic receiving 9361 citation(s).

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TL;DR: The fast generation of discrete random variables with arbitrary frequency distributions is discussed, related to rejection techniques but differs from them in that all samples comprising the input data contribute to the samples in the target distribution.

Abstract: The fast generation of discrete random variables with arbitrary frequency distributions is discussed. The proposed method is related to rejection techniques but differs from them in that all samples comprising the input data contribute to the samples in the target distribution. The software implementation of the method requires at most two memory references and a comparison. The method features good accuracy and modest storage requirements. I t is particularly useful in small computers with limited memory capacity.

504 citations

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^{1}TL;DR: This paper presents a version of the Fourier-series method for numerically inverting probability generating functions with a simple algorithm with a convenient error bound from the discrete Poision summation formula.

Abstract: Random quantities of interest in operations research models can often be determined conveniently in the form of transforms. Hence, numerical transform inversion can be an effective way to obtain desired numerical values of cumulative distribution functions, probability density functions and probability mass functions. However, numerical transform inversion has not been widely used. This lack of use seems to be due, at least in part, to good simple numerical inversion algorithms not being well known. To help remedy this situation, in this paper we present a version of the Fourier-series method for numerically inverting probability generating functions. We obtain a simple algorithm with a convenient error bound from the discrete Poision summation formula. The same general approach applies to other transforms.

241 citations

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225 citations

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174 citations

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TL;DR: The statistical distribution of capture times is obtained from Monte Carlo calculations and shows a crossover from power-law to exponential behavior, and predicts the distribution function for a lattice with perfect mixing.

Abstract: Reaction kinetics in a cell or cell membrane is modeled in terms of the first passage time for a random walker at a random initial position to reach an immobile target site in the presence of a hierarchy of nonreactive binding sites. Monte Carlo calculations are carried out for the triangular, square, and cubic lattices. The mean capture time is expressed as the product of three factors: the analytical expression of Montroll for the capture time in a system with a single target and no binding sites; an exact expression for the mean escape time from the set of lattice points; and a correction factor for the number of targets present. The correction factor, obtained from Monte Carlo calculations, is between one and two. Trapping may contribute significantly to noise in reaction rates. The statistical distribution of capture times is obtained from Monte Carlo calculations and shows a crossover from power-law to exponential behavior. The distribution is analyzed using probability generating functions; this analysis resolves the contributions of the different sources of randomness to the distribution of capture times. This analysis predicts the distribution function for a lattice with perfect mixing; deviations reflect imperfect mixing in an ordinary random walk.

148 citations