scholarly journals The Distribution of the Ratio of Two Quadratic Forms

1955 ◽  
Vol 8 (3) ◽  
pp. 402 ◽  
Author(s):  
GS Watson
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Quadratic Forms ◽  
Multivariate Normal Distribution ◽  
Linear Functions ◽  
Multivariate Normal ◽  
Exact Probability ◽  
Exact Probability Distribution ◽  
Integral Order

The exact probability distribution of the ratio of two quadratic forms is given in the case where the quadratic forms and the multivariate normal distribution of the variables is such that the ratio is a ratio of linear functions of gamma variables of integral order.

scholarly journals Modeling parameters of structured clays as a multivariate normal distribution

2012 ◽  
Vol 49 (5) ◽  
pp. 522-545 ◽  
Author(s):  
Jianye Ching ◽  
Kok-Kwang Phoon
Keyword(s):  
Shear Strength ◽  
Probability Distribution ◽  
Normal Distribution ◽  
Multivariate Normal Distribution ◽  
Undrained Shear Strength ◽  
Multivariate Distribution ◽  
Soil Parameters ◽  
Multivariate Normal ◽  
Multivariate Probability Distribution ◽  
Undrained Shear

This study explores the possibility of modeling liquidity index, undrained shear strength, remolded undrained shear strength, preconsolidation stress, and vertical effective stress of structured clays (sensitive or quick clays) as a multivariate normal distribution. The literature is replete with correlation equations between two soil parameters. Consistent synthesis of more than two soil parameters through construction of a multivariate probability distribution function is rare, despite obvious practical usefulness of such an approach. This study compiles a large database of structured clays to construct the multivariate probability distribution among the aforementioned five soil parameters. This multivariate distribution is then used to simulate the correlations between soil parameters of interest and to derive useful equations for Bayesian inference. This constructed multivariate distribution and equations are further validated by another independent database of structured clays as well as by empirical equations proposed in the literature.

scholarly journals Blinking statistics and molecular counting in direct stochastic reconstruction microscopy (dSTORM)

2021 ◽  
Author(s):  
Lekha Patel ◽  
David Williamson ◽  
Dylan M Owen ◽  
Edward A K Cohen
Keyword(s):  
Probability Distribution ◽  
Single Molecule ◽  
Immunological Synapse ◽  
Training Data ◽  
Supplementary Information ◽  
Cellular Structures ◽  
Exact Probability ◽  
Stochastic Optical Reconstruction Microscopy ◽  
Exact Probability Distribution ◽  
Optical Reconstruction

Abstract Motivation Many recent advancements in single-molecule localization microscopy exploit the stochastic photoswitching of fluorophores to reveal complex cellular structures beyond the classical diffraction limit. However, this same stochasticity makes counting the number of molecules to high precision extremely challenging, preventing key insight into the cellular structures and processes under observation. Results Modelling the photoswitching behaviour of a fluorophore as an unobserved continuous time Markov process transitioning between a single fluorescent and multiple dark states, and fully mitigating for missed blinks and false positives, we present a method for computing the exact probability distribution for the number of observed localizations from a single photoswitching fluorophore. This is then extended to provide the probability distribution for the number of localizations in a direct stochastic optical reconstruction microscopy experiment involving an arbitrary number of molecules. We demonstrate that when training data are available to estimate photoswitching rates, the unknown number of molecules can be accurately recovered from the posterior mode of the number of molecules given the number of localizations. Finally, we demonstrate the method on experimental data by quantifying the number of adapter protein linker for activation of T cells on the cell surface of the T-cell immunological synapse. Availability and implementation Software and data available at https://github.com/lp1611/mol_count_dstorm. Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals On the birthday problem: some generalizations and applications

2003 ◽  
Vol 2003 (60) ◽  
pp. 3827-3840 ◽  
Author(s):  
P. N. Rathie ◽  
P. Zörnig
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Exact Probability ◽  
Confluent Hypergeometric Functions ◽  
Birthday Problem ◽  
Exact Probability Distribution

We study the birthday problem and some possible extensions. We discuss the unimodality of the corresponding exact probability distribution and express the moments and generating functions by means of confluent hypergeometric functionsU(−;−;−)which are computable using the software Mathematica. The distribution is generalized in two possible directions, one of them consists in considering a random graph with a single attracting center. Possible applications are also indicated.

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