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: It is shown that one of the diffusing diffusivity models is equivalent to Brownian motion in the presence of a sink and a class of models for which it is possible to find analytical solutions is introduced.
Abstract: It has been found in many experiments that the mean square displacement of a Brownian particle x(T) diffusing in a rearranging environment is strictly Fickian, obeying ⟨(x(T))2⟩ ∝ T, but the probability distribution function for the displacement is not Gaussian. An explanation of this is that the diffusivity of the particle itself is changing as a function of time. Models for this diffusing diffusivity have been solved analytically in the limit of small time, but simulations were necessary for intermediate and large times. We show that one of the diffusing diffusivity models is equivalent to Brownian motion in the presence of a sink and introduce a class of models for which it is possible to find analytical solutions. Our solution gives ⟨(x(T))2⟩ ∝ T for all times and at short times the probability distribution function of the displacement is exponential which crosses over to a Gaussian in the limit of long times and large displacements.
113 citations
TL;DR: The cumulative distribution function, the probability density function and the moment generating function of the MRT output signal-to-noise ratio (SNR) with imperfect CSI are derived, enabling the evaluation of some useful performance metrics such as the average error rate and the outage performance.
Abstract: Maximal ratio transmission (MRT) is designed assuming the availability of perfect channel state information (CSI) at both the transmitter and the receiver. However, perfect CSI is not available in practice. This paper investigates the impact of Gaussian estimation errors on the MRT performance in independently and identically distributed (i.i.d.) Rayleigh fading channels. We derive the cumulative distribution function (cdf), the probability density function (pdf) and the moment generating function (mgf) of the MRT output signal-to-noise ratio (SNR) with imperfect CSI, enabling the evaluation of some useful performance metrics such as the average error rate and the outage performance. Numerical and simulation results are provided to show the impact of imperfect CSI on the MRT performance.
113 citations
TL;DR: In this paper, a second-order reliability method for reliability analysis is presented using saddlepoint approximation, which is obtained by the secondorder Taylor series expansion at the most probable point.
113 citations
TL;DR: In this article, the authors generate skew probability density functions (pdfs) of the form 2f(u)G(λu), where f is taken to be a normal pdf while the cumulative distributive function G is taken from one of normal, Student's t, Cauchy, Laplace, logistic or uniform distribution.
112 citations
TL;DR: In this paper, a unified approach to the construction of confidence bands in nonparametric density estimation and regression is described, based on interpolation formulae in numerical differentiation, and their arguments generate a variety of bands depending on the assumptions one is prepared to make about derivatives of the unknown function.
112 citations