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.

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.

THE DISTRIBUTION OF DISTANCES BETWEEN RANDOMLY CONSTRUCTED GENOMES: GENERATING FUNCTION, EXPECTATION, VARIANCE AND LIMITS

2008 ◽  
Vol 06 (01) ◽  
pp. 23-36 ◽  
Author(s):  
WEI XU
Keyword(s):  
Probability Distribution ◽  
Generating Function ◽  
Chromosomal Rearrangement ◽  
Genomic Distance ◽  
Exact Probability ◽  
Breakpoint Graph ◽  
Exact Probability Distribution ◽  
Number Of Cycles ◽  
Linear Chromosomes ◽  
Large Repertoire

Based on a large repertoire of chromosomal rearrangement operations, the genomic distance d between two genomes with χr and χb linear chromosomes, respectively, both containing the same (or orthologous) n genes or markers, is d = n + max (χr,χb) - c, where c is the number of cycles in the breakpoint graph of the two genomes. In this paper, we study the exact probability distribution of c. We derive the expectation and variance, and show that, in the limit, the expectation of d is [Formula: see text].

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