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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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Journal ArticleDOI
TL;DR: The approach of finding the asymptotic distributions of many run statistics via the theorems established for m-dependent sequence of random variables is based on the construction of runs in a sequence of m- dependent random variables.

7 citations

Journal ArticleDOI
TL;DR: The random variable Z appears in several domains of applications and can be easily proved by first noticing that for vI < 1, 1 V2 co, the theorem can be easy proved.
Abstract: Communicated by Jerzy Neyman, August 1, 1966 1. All random variables considered in this note are nonnegative and integervalued. Generically, the probability generating functions (P. G. F.) of such variables are denoted by G with subscripts identifying the random variables concerned. The argument, or arguments of the P. G. F., denoted by either u or v, will be assumed to have their moduli less than unity. The note gives a simple formula for the P. G. F. of the absolute difference Z = i X2|, where X1 and X2 are arbitrary random variables of the kind considered. Later, this formula is used to obtain a characterization of the geometric distribution. The random variable Z appears in several domains of applications. See, for instance, David.1 2. THEOREM 1. Whatever be the random variables X1 and X2, we have = 1 (2I 1 -V2 *GS x(eiGe i)dO. (1) i(V) 2~r Jo 1+ v2-2v cosO 1, The theorem can be easily proved by first noticing that for vI < 1, 1 V2 co

7 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied the asymptotic behavior of the probability of extinction of a critical branching process Z at moment $n\rightarrow \infty,$ and showed that if the logarithm of the (random) expectation of the offspring number of a particle belongs to the domain of attraction of a non-Gaussian stable law, then the extinction occurs at time moment T owing to a very unfavorable environment forcing the process, having an exponentially large population, to die out instantly.
Abstract: Let T be the extinction moment of a critical branching process $Z=(Z_{n},n\ge 0)$ in a random environment specified by independent identically distributed probability generating functions. We study the asymptotic behavior of the probability of extinction of the process Z at moment $n\rightarrow \infty ,$ and show that if the logarithm of the (random) expectation of the offspring number of a particle belongs to the domain of attraction of a non-Gaussian stable law, then the extinction occurs at time moment T owing to a very unfavorable environment forcing the process, having at time moment $T-1$ an exponentially large population, to die out instantly. We also give an interpretation of the obtained results in terms of random walks in a random environment.

7 citations

Journal ArticleDOI
TL;DR: In this article, the principal resonance of the Duffing oscillator to combined deterministic and random external excitation was investigated and the theoretical analysis were verified by numerical results using the double peak probability density function.
Abstract: The principal resonance of Duffing oscillator to combined deterministic and random external excitation was investigated. The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.

7 citations

Patent
10 Aug 2007
TL;DR: In this paper, a probability density function separator for separating a predetermined component from a given probability density functions is defined, where a domain converting section is used to convert the given density function into the spectrum of a frequency domain, and a fixed component calculating section for multiplying a first null frequency of the spectrum in the frequency domain by a multiplier coefficient corresponding to the type of the distribution of the fixed component included in the given distribution.
Abstract: A probability density function separator for separating a predetermined component from a given probability density function comprises: a domain converting section for converting the given probability density function into the spectrum of a frequency domain; and a fixed component calculating section for multiplying a first null frequency of the spectrum of the frequency domain by a multiplier coefficient corresponding to the type of the distribution of a fixed component included in the given probability density function so as to calculate the peak-to-peak value of the probability density function of the fixed component

7 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20236
202211
20217
202014
201912
20188