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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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Journal ArticleDOI
TL;DR: In this article, a point estimate method is proposed for computing the first four statistical moments of structural response which is a function of input random variables, such as material properties, geometrical parameters and loading conditions, on the structural responses.
Abstract: Structural probabilistic analysis quantifies the effect of input random variables, such as material properties, geometrical parameters and loading conditions, on the structural responses. The point estimate method (PEM) is a direct and easy-used way to perform the structural probabilistic analysis in practice. In this paper, a novel and efficient point estimate method is proposed for computing the first four statistical moments of structural response which is a function of input random variables. The method adopts Nataf transformation to replace Rosenblatt transformation in conventional point estimate method. Because of the nature of engineering problems and limited statistical data, the joint probability density function (PDF) of all input random variables is hard to acquire, but it must be known in Rosenblatt transformation. A more common case is that the marginal PDF of each random variable and the correlation matrix are available, which just satisfy the service condition of Nataf transformation. Hence the Nataf transformation based point estimate method is particularly suitable for engineering applications. The comparison between the proposed method and the conventional point estimate method shows that (1) they are equivalent when all random variables are mutually independent; (2) if the marginal PDFs and the correlation matrix are known, the conventional PEM cannot be applicable, but the proposed method can give a rational approximation. Finally, the procedure is demonstrated in detail through a simple illustration.

140 citations

Journal ArticleDOI
TL;DR: This paper proposes a generalized probability density function based on the nth power of a cosine-squared function that derives the average covariance matrix for various different elementary scatterers and shows that the result has a very simple analytical form suitable for use in model-based decomposition schemes.
Abstract: Current polarimetric model-based decomposition techniques are limited to specific types of vegetation because of their assumptions about the volume scattering component. In this paper, we propose a generalized probability density function based on the nth power of a cosine-squared function. This distribution is completely characterized by two parameters; a mean orientation angle and the power of the cosine-squared function. We show that the underlying randomness of the distribution is only a function of the power of the cosine-squared function. We then derive the average covariance matrix for various different elementary scatterers showing that the result has a very simple analytical form suitable for use in model-based decomposition schemes.

140 citations

Proceedings ArticleDOI
Kai-ching Chu1
01 Dec 1972
TL;DR: It is shown in this paper that this class of densities can be expressed as integrals of a set of Gaussian densities and it is proved that the conditional expectation is linear with exactly the same form as the Gaussian case.
Abstract: A random variable is said to have elliptical distribution if its probability density is a function of a quadratic form. This class includes the Gaussian and many other useful densities in statistics. It is shown in this paper that this class of densities can be expressed as integrals of a set of Gaussian densities. This property is not changed under linear transformation of the random variables. It is also proved in this paper that the conditional expectation is linear with exactly the same form as the Gaussian case. Many estimation results of the Gaussian case can be readily extended. Problems of computing optimal estimation, filtering, stochastic control, and team decisions in various linear systems become tractable for this class of random processes.

140 citations

Journal ArticleDOI
01 Jan 2005
TL;DR: In this article, an uncertainty analysis method is proposed with the purpose of accurately and efficiently estimating the cumulative distribution function (CDF), probability density function (PDF), and statistical moments of a response given the distributions of input variables.
Abstract: Uncertainty analysis, which assesses the impact of the uncertainty of input variables on responses, is an indispensable component in engineering design under uncertainty, such as reliability-based design and robust design. However, uncertainty analysis is an unaffordable computational burden in many engineering problems. In this paper, an uncertainty analysis method is proposed with the purpose of accurately and efficiently estimating the cumulative distribution function (CDF), probability density function (PDF), and statistical moments of a response given the distributions of input variables. The bivariate dimension reduction method and numerical integration are used to calculate the moments of the response; then saddlepoint approximations are employed to estimate the CDF and PDF of the response. The proposed method requires neither the derivatives of the response nor the search of the most probable point, which is needed in the commonly used first and second order reliability methods (FORM and SORM) and the recently developed first order saddlepoint approximation. The efficiency and accuracy of the proposed method is illustrated with three example problems. With the same computational cost, this method is more accurate for reliability assessment and much more efficient for estimating the full range of the distribution of a response than FORM and SORM. This method provides results as accurate as Monte Carlo simulation, with significantly reduced computational effort.

140 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