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, a new fractional Langevin-type stochastic dierential equation is introduced, which is derived from the standard Langevin equation, by replacing the rst-order derivative with respect to time by the fractional derivative of order ; and by replacing white noise" Gaussian force by the generalized shot noise", each pulse of which has a random amplitude with the -stable Levy distribution.
Abstract: A new extension of a fractality concept in nancial mathematics has been developed. We have introduced a new fractional Langevin-type stochastic dierential equation that diers from the standard Langevin equation: (i) by replacing the rst-order derivative with respect to time by the fractional derivative of order ; and (ii) by replacing \white noise" Gaussian stochastic force by the generalized \shot noise", each pulse of which has a random amplitude with the -stable Levy distribution. As an application of the developed fractional non-Gaussian dynamical approach the expression for the probability distribution function (pdf) of the returns has been established. It is shown that the obtained fractional pdf ts well the central part and the tails of the empirical distribution of S&P 500 returns. c 2000 Elsevier Science B.V. All rights reserved.
394 citations
TL;DR: It is shown that as the temperature approaches zero, the algorithm becomes the basic ISODATA algorithm and the method is independent of the initial choice of cluster means.
393 citations
TL;DR: In this article, the authors considered the convergence of variance type models for a regression function or for the logarithm of a probability function, conditional probability functions, density function, hazard function, or spectral density function.
Abstract: Analysis of variance type models are considered for a regression function or for the logarithm of a probability function, conditional probability function, density function, conditional density function, hazard function, conditional hazard function or spectral density function. Polynomial splines are used to model the main effects, and their tensor products are used to model any interaction components that are included. In the special context of survival analysis, the baseline hazard function is modeled and nonproportionality is allowed. In general, the theory involves the $L_2$ rate of convergence for the fitted model and its components. The methodology involves least squares and maximum likelihood estimation, stepwise addition of basis functions using Rao statistics, stepwise deletion using Wald statistics and model selection using the Bayesian information criterion, cross-validation or an independent test set. Publicly available software, written in C and interfaced to S/S-PLUS, is used to apply this methodology to real data.
387 citations
TL;DR: In this article, the distribution and density functions of the ratio of two normal random variables were studied in terms of the vivariate normal distribution and the Nicholson's V function, both of which have been extensively studied and for which tables and computational procedures are readily available.
Abstract: The principal part of this paper is devoted to the study of the distribution and density functions of the ratio of two normal random variables. It gives several representations of the distribution function in terms of the vivariate normal distribution and Nicholson's V function, both of which have been extensively studied, and for which tables and computational procedures are readily available. One of these representations leads to an easy derivation of the density function in terms of the Cauchy density and the normal density and integral. A number of graphs of the possible shapes of the density are given, together with an indication of when the density is unimodal or bimodal. The last part of the paper discusses the distribution of the ratio (u 1+ ¨˙ +un )/(v 1+ ¨˙ +vm ) where the u's and v's are independent, uniform variables. The exact distribution for all n and m is given, and some approximations discussed.
386 citations
TL;DR: A method to estimate the probability that a conflict will occur, given a pair of predicted trajectories and their levels of uncertainty is presented.
Abstract: The safety and efficiency of free flight will benefit from automated conflict prediction and resolution advisories. Conflict prediction is based on trajectory prediction and is less certain the farther in advance the prediction, however. An estimate is therefore needed of the probability that a conflict will occur, given a pair of predicted trajectories and their levels of uncertainty. This paper presents a method to estimate that conflict probability. The trajectory prediction errors are modeled as normally distributed, and the two error covariances for an aircraft pair are combined into a single, equivalent covariance of the relative position. A coordinate transformation is then used to derive an analytical solution. Numerical examples and a Monte Carlo validation are presented. (Author)
384 citations