Approximate Probability Distributions for the Extreme Spread

1975 ◽  
Author(s):  
M. S. Taylor ◽  
F. E. Grubbs
Keyword(s):  
Probability Distributions ◽  
Approximate Probability

scholarly journals The Sum and Difference of Two Lognormal Random Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
C. F. Lo
Keyword(s):  
Stochastic Process ◽  
Probability Distributions ◽  
Operator Splitting ◽  
Splitting Method ◽  
Unified Approach ◽  
New Approach ◽  
Numerical Examples ◽  
Operator Splitting Method ◽  
Approximate Probability ◽  
Analytical Series

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum ofNlognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

scholarly journals Bin-Free Fitting of Probability Distributions Emphasizing DNA Histograms

2017 ◽  
Author(s):  
Nash Rochman
Keyword(s):  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Cumulative Distribution ◽  
Data Set ◽  
Cell Synchronization ◽  
Approximate Probability ◽  
A Cell ◽  
The Right ◽  
Probability Density Function Pdf ◽  
Dna Histograms

AbstractIt is often challenging to find the right bin size when constructing a histogram to represent a noisy experimental data set. This problem is frequently faced when assessing whether a cell synchronization experiment was successful or not. In this case the goal is to determine whether the DNA content is best represented by a unimodal, indicating successful synchronization, or bimodal, indicating unsuccessful synchronization, distribution. This choice of bin size can greatly affect the interpretation of the results; however, it can be avoided by fitting the data to a cumulative distribution function (CDF). Fitting data to a CDF removes the need for bin size selection. The sorted data can also be used to reconstruct an approximate probability density function (PDF) without selecting a bin size. A simple CDF-based approach is presented and the benefits and drawbacks relative to usual methods are discussed.

scholarly journals Machine-learning inference of the interior structure of low-mass exoplanets

2020 ◽  
Author(s):  
Philipp Baumeister ◽  
Sebastiano Padovan ◽  
Nicola Tosi ◽  
Grégoire Montavon ◽  
Nadine Nettelmann ◽  
...  
Keyword(s):  
Machine Learning ◽  
Neural Networks ◽  
Probability Distributions ◽  
Mixture Distribution ◽  
Gaussian Mixture ◽  
Love Number ◽  
Mixture Density ◽  
Single Output ◽  
Approximate Probability ◽  
Low Mass

<p>We explore the application of machine-learning, based on mixture density neural networks (MDNs), to the interior characterization of low-mass exoplanets up to 25 Earth masses constrained by mass, radius, and fluid Love number k<sub>2</sub>. MDNs are a special subset of neural networks, able to predict the parameters of a Gaussian mixture distribution instead of single output values, which enables them to learn and approximate probability distributions. With a dataset of 900,000 synthetic planets, consisting of an iron-rich core, a silicate mantle, a high-pressure ice shell, and a gaseous H/He envelope, we train an MDN using planetary mass and radius as inputs to the network. We show that the MDN is able to infer the distribution of possible thicknesses of each planetary layer from mass and radius of the planet. This approach obviates the time-consuming task of calculating such distributions with a dedicated set of forward models for each individual planet.</p><p>The fluid Love number k<sub>2</sub> bears constraints on the mass distribution in the planets' interior and will be measured for an increasing number of exoplanets in the future. Adding k<sub>2</sub> as an input to the MDN significantly decreases the degeneracy of possible interior structures.</p>

On Randomly Excited Hysteretic Structures

1990 ◽  
Vol 57 (2) ◽  
pp. 442-448 ◽  
Author(s):  
G. Q. Cai ◽  
Y. K. Lin
Keyword(s):  
Energy Dissipation ◽  
Narrow Band ◽  
Probability Distributions ◽  
Gaussian White Noise ◽  
Energy Balancing ◽  
Dissipation Energy ◽  
System Parameters ◽  
Hysteretic Systems ◽  
Approximate Probability ◽  
Simulation Results

Approximate probability distributions of certain response variables are obtained for hysteretic systems under Gaussian white-noise excitations. The approximate method used is a generalization of a dissipation-energy-balancing procedure, developed previously for nonlinear but basically nonhysteretic systems. Some new issues related particularly to hysteresis models are explained and resolved. The method is applicable to either bilinear or smooth-type hysteresis without the restriction that the response be a narrow-band process or the energy dissipation be small. Comparison of computed results with available simulation results indicates that the proposed method is accurate for wide ranges of excitation levels and system parameters.

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