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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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Book ChapterDOI
TL;DR: In this article, the Boltzmann formula for lower temperatures has been developed for a correction term, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.
Abstract: The probability of a configuration is given in classical theory by the Boltzmann formula exp [— V/hT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.

5,865 citations

Journal ArticleDOI
TL;DR: In this article, some aspects of the estimation of the density function of a univariate probability distribution are discussed, and the asymptotic mean square error of a particular class of estimates is evaluated.
Abstract: This note discusses some aspects of the estimation of the density function of a univariate probability distribution. All estimates of the density function satisfying relatively mild conditions are shown to be biased. The asymptotic mean square error of a particular class of estimates is evaluated.

4,284 citations

Journal ArticleDOI
TL;DR: In this article, various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable are described. But none of these methods can be used for shape optimization in a general setting.
Abstract: Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.

3,434 citations

Journal ArticleDOI
TL;DR: Krystek as discussed by the authors provides a comprehensive and self-contained overview of random data analysis, including derivations of the key relationships in probability and random-process theory not usually found to such extent in a book of this kind.
Abstract: This is a new edition of a book on random data analysis which has been on the market since 1966 and which was extensively revised in 1971. The book has been a bestseller since. It has been fully updated to cover new procedures developed in the last 15 years and extends the discussion to a broad range of applied fields, such as aerospace, automotive industries or biomedical research. The primary purpose of this book is to provide a practical reference and tool for working engineers and scientists investigating dynamic data or using statistical methods to solve engineering problems. It is comprehensive and self-contained and expands the coverage of the theory, including derivations of the key relationships in probability and random-process theory not usually found to such extent in a book of this kind. It could well be used as a teaching textbook for advanced courses on the analysis of random processes. The first four chapters present the background material on descriptions of data, properties of linear systems and statistical principles. They also include probability distribution formulas for one-, two- and higher-order changes of variables. Chapter five gives a comprehensive discussion of stationary random-process theory, including material on wave-number spectra, level crossings and peak values of normally distributed random data. Chapters six and seven develop mathematical relationships for the detailed analysis of single input/output and multiple input/output linear systems including algorithms. In chapters eight and nine important practical formulas to determine statistical errors in estimates of random data parameters and linear system properties from measured data are derived. Chapter ten deals with data aquisition and processing, including data qualification. Chapter eleven describes methods of data analysis such as data preparation, Fourier transforms, probability density functions, auto- and cross-correlation, spectral functions, joint record functions and multiple input/output functions. Chapter twelve shows how to handle nonstationary data analysis, classification of nonstationary data, probability structure of nonstationary data, calculation of nonstationary mean values or mean square values, correlation structures of nonstationary data and spectral structures of nonstationary data. The last chapter deals with the Hilbert transform including applications for both nondispersive and dispersive propagation problems. All chapters include many illustrations and references as well as examples and problem sets. This allows the reader to use the book for private study purposes. Altogether the book can be recommended for practical working engineers and scientists to support their daily work, as well as for university readers as a teaching textbook in advanced courses. M Krystek

3,390 citations

MonographDOI
01 Jan 1972
TL;DR: In this article, the probability density, Fourier transforms and characteristic functions, joint statistics and statistical independence, Correlation functions and spectra, the central limit theorem, and the relation functions are discussed.
Abstract: This chapter contains sections titled: The probability density, Fourier transforms and characteristic functions, Joint statistics and statistical independence, Correlation functions and spectra, The central limit theorem

3,260 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023382
2022906
2021906
20201,047
20191,117
20181,083