Random Variables, Random Vectors, and Distribution Functions

2005 ◽  
pp. 16-32
Keyword(s):  
Random Variables ◽  
Distribution Functions ◽  
Random Vectors

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

scholarly journals Finding Maximum Clique in Stochastic Graphs Using Distributed Learning Automata

2015 ◽  
Vol 23 (01) ◽  
pp. 1-31 ◽  
Author(s):  
Alireza Rezvanian ◽  
Mohammad Reza Meybodi
Keyword(s):  
Community Structure ◽  
Dominating Set ◽  
Learning Automata ◽  
Random Variables ◽  
Graph Model ◽  
Maximum Clique ◽  
Distribution Functions ◽  
Maximum Clique Problem ◽  
Stochastic Graphs ◽  
Stochastic Graph

Because of unpredictable, uncertain and time-varying nature of real networks it seems that stochastic graphs, in which weights associated to the edges are random variables, may be a better candidate as a graph model for real world networks. Once the graph model is chosen to be a stochastic graph, every feature of the graph such as path, clique, spanning tree and dominating set, to mention a few, should be treated as a stochastic feature. For example, choosing stochastic graph as the graph model of an online social network and defining community structure in terms of clique, and the associations among the individuals within the community as random variables, the concept of stochastic clique may be used to study community structure properties. In this paper maximum clique in stochastic graph is first defined and then several learning automata-based algorithms are proposed for solving maximum clique problem in stochastic graph where the probability distribution functions of the weights associated with the edges of the graph are unknown. It is shown that by a proper choice of the parameters of the proposed algorithms, one can make the probability of finding maximum clique in stochastic graph as close to unity as possible. Experimental results show that the proposed algorithms significantly reduce the number of samples needed to be taken from the edges of the stochastic graph as compared to the number of samples needed by standard sampling method at a given confidence level.

scholarly journals Stochastic Single Footfall Trace Model for Pedestrian Walking Load

2019 ◽  
Vol 19 (03) ◽  
pp. 1950029 ◽  
Author(s):  
Jun Chen ◽  
Guo Ding ◽  
Stana Živanović
Keyword(s):  
Random Variables ◽  
Distribution Functions ◽  
Dynamic Force ◽  
Gait Parameters ◽  
Proposed Model ◽  
Vibration Serviceability ◽  
Reliable Model ◽  
Time Histories ◽  
Trace Model ◽  
Force Time

Developing a model for the dynamic force generated by a pedestrian’s foot on a supporting structure (single footfall trace model) is crucial to advanced numerical analysis and vibration serviceability assessment of the structure. A reliable model needs to reflect the inter-subject and intra-subject randomness of human walking. This paper introduces a stochastic single footfall trace model in the form of a Fourier series in which body weight, walking frequency, and the first eight harmonics are treated as random variables. An experiment used 73 test subjects, walking at a range of pacing frequencies, to record force time histories and the corresponding gait parameters. Two variability descriptors were used to indicate inter-subject and intra-subject randomness. Further statistical analysis identified the relationships between key parameters as well as the probability distribution functions of random variables. In the final step, an application of the proposed single footfall trace model was developed and tested. The proposed model represented the experimental data well in both time and frequency domains.

scholarly journals Transformation of circular random variables based on circular distribution functions

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 5931-5947
Author(s):  
Hatami Mojtaba ◽  
Alamatsaz Hossein
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Real Data ◽  
Random Variable ◽  
Distribution Functions ◽  
Likelihood Method ◽  
Circular Distribution ◽  
Data Set ◽  
Trigonometric Moments ◽  
Von Mises

In this paper, we propose a new transformation of circular random variables based on circular distribution functions, which we shall call inverse distribution function (id f ) transformation. We show that M?bius transformation is a special case of our id f transformation. Very general results are provided for the properties of the proposed family of id f transformations, including their trigonometric moments, maximum entropy, random variate generation, finite mixture and modality properties. In particular, we shall focus our attention on a subfamily of the general family when id f transformation is based on the cardioid circular distribution function. Modality and shape properties are investigated for this subfamily. In addition, we obtain further statistical properties for the resulting distribution by applying the id f transformation to a random variable following a von Mises distribution. In fact, we shall introduce the Cardioid-von Mises (CvM) distribution and estimate its parameters by the maximum likelihood method. Finally, an application of CvM family and its inferential methods are illustrated using a real data set containing times of gun crimes in Pittsburgh, Pennsylvania.

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