Topic
Probability density function
About: Probability density function is a research topic. Over the lifetime, 22321 publications have been published within this topic receiving 422885 citations. The topic is also known as: probability function & PDF.
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TL;DR: In this paper, it was shown that the probability of collision induced rotational transfer (RT) is controlled by the factors that control the angular momentum (AM) change, and that such a calculation leads to an exponential fall of RT probabilities with transferred AM, a consequence of the radial dependence of the repulsive part of the intermolecular potential.
Abstract: We have re‐examined critical experiments on collision induced rotational transfer (RT) and conclude that the probability of RT is controlled by the factors that control the probability of angular momentum (AM) change. The probability of energy change seems less important in this respect. In the light of this we suggest a model for RT in which the probability of AM change is calculated directly and present a formalism for this purpose. We demonstrate that such a calculation leads to an exponential‐like fall of RT probabilities with transferred AM, a consequence of the radial dependence of the repulsive part of the intermolecular potential. Thus in this AM model, the exponential gap law has a simple physical origin. The AM model we describe may be used as the basis of an inversion routine through which it is possible to convert RT data into a probability density of the repulsive anisotropy. Through this model therefore it is possible to relate experimental RT data directly to the forces that are responsible for rotational transfer. The hard ellipse model is used in this work to relate calculated anisotropies to a form that includes an isotropic component. The result is a representation of the intermolecular potential through which new insights into the RT process are gained.
98 citations
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TL;DR: In this paper, the Weibull-gamma (WG) distribution is used for modeling fading environments when multipath is superimposed on shadowing, and the probability density, cumulative distribution, characteristic functions and the moments are derived in closed form.
Abstract: The Weibull-gamma (WG) distribution, which is appropriate for modelling fading environments when multipath is superimposed on shadowing, is introduced and studied. For this composite distribution the probability density, cumulative distribution, characteristic functions and the moments are derived in closed form. Furthermore, the average bit error and outage probabilities of a receiver operating over WG fading channels are assessed and compared with the corresponding performances obtained using other composite distributions.
98 citations
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TL;DR: In this article, a new class of continuous distributions with two extra shape parameters named the generalized odd log-logistic family of distributions was proposed, which can be expressed as a linear combination of exponentiated densities based on the same baseline distribution.
Abstract: We propose a new class of continuous distributions with two extra shape parameters named the generalized odd log-logistic family of distributions. The proposed family contains as special cases the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, two entropy measures and order statistics are obtained. We derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behaviour of the estimators by means of Monte Carlo simulations. We introduce the log-odd log-logistic Weibull regression model with censored data based on the odd log-logistic-Weibull distribution. The importance of the new family is illustrated using three real data sets. These applications indicate that this family can...
98 citations
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TL;DR: The normalized scattering coefficient is obtained whose probability density function is predicted to be universal in that it depends only on the loss of the cavity, and which applies to scattering measurements on any wave chaotic system.
Abstract: We consider the statistics of the scattering coefficient S of a chaotic microwave cavity coupled to a single port. We remove the nonuniversal effects of the coupling from the experimental S data using the radiation impedance obtained directly from the experiments. We thus obtain the normalized scattering coefficient whose probability density function (PDF) is predicted to be universal in that it depends only on the loss (quality factor) of the cavity. We compare experimental PDFs of the normalized scattering coefficients with those obtained from random matrix theory (RMT), and find excellent agreement. The results apply to scattering measurements on any wave chaotic system.
98 citations
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TL;DR: In this article, the beta Frechet (BF) distribution is expressed as linear combinations of the exponentiated Frechet and Frechet density functions, and the moments and order statistics are derived.
Abstract: Nadarajah and Gupta (2004) introduced the beta Frechet (BF) distribution, which is a generalization of the exponentiated Frechet (EF) and Frechet distributions, and obtained the probability density and cumulative distribution functions. However, they did not investigate the moments and the order statistics. In this article, the BF density function and the density function of the order statistics are expressed as linear combinations of Frechet density functions. This is important to obtain some mathematical properties of the BF distribution in terms of the corresponding properties of the Frechet distribution. We derive explicit expansions for the ordinary moments and L-moments and obtain the order statistics and their moments. We also discuss maximum likelihood estimation and calculate the information matrix which was not given in the literature. The information matrix is numerically determined. The usefulness of the BF distribution is illustrated through two applications to real data sets.
98 citations