Applied Probability and Statistics

This research determined time to failure rate and number of successful transaction of selected banks in Nigeria, using Log normal distribution. Transformation technique was applied to the log-normal model to obtain a quadratic equation or polynomial regression that assisted in determining the parameters of the log-normal model. In addition, one-way ANOVA was used to test for equality of the average (or mean) time to failure rate and average number of successful service time of the banks. The research fitted the log-normal models of the banks with the help of SPSS 21 statistical software and the result showed that GT-Bank model has the highest variation of 90.3% for number of successful service time (t), while Fidelity bank model has the highest variation of 56.6% for time of failure rate. The one-way ANOVA result of the number of successful service time (min) showed a significant difference. The Tukey comparison tests showed that GT bank is significant at 5% and 10% from other banks. Hence, the number of successful service time (min) were not the same for all the five banks. However, the one-way ANOVA result of the banks in term of number of Time to Failure (t) (min) showed no significant difference among the five banks.

The theory of dual numerical means of random and experienced variables is briefly described in the framework of the new theory of experience and the chance that arises as an axiomatic synthesis of two dual theories — the Kolmogorov theory of probability and the theory of believability. A new term is introduced for the numerical mean of the experienced variable — mathematical reflection, which is dual to the mathematical expectation of a random variable within the framework of the new theory. The basic properties and examples of dual numerical means are considered.

Logistic regression analysis of factors contributing to bank deposit subscription and direct marketing success.

View project at:

https://github.com/JackKRasmus-Vorrath/Logistic_Regression_Direct-Marketing_Bank-Term-Deposit-Subscription

Example 1. Consider the two-state, continuous-time Markov process with transition rate diagram for some positive constants A and B. The generator matrix is given by Q = −A A B −B. Solve the forward Kolmogorov equation for a given initial distribution π(0). Solution. The equation for π 1 (t) is ∂π 1 (t) ∂t = −Aπ 1 (t) + Bπ 2 (t); π 1 (0) given. But π 1 (t) = 1 − π 2 (t) so ∂π 1 (t) ∂t = −(A + B)π 1 (t) + B; π 1 (0) given. By differentiation we check that this equation has the solution π t (t) = π 1 (0)e −(A+B)t + B A + B (1 − e −(A+B)t) so that π(t) = π(0)e −(A+B)t + B A + B , A A + B (1 − e −(A+B)t). For any initial distribution π(0) lim t→∞ π(t) = B A + B , A A + B. The rate of convergence is exponential, with rate parameter A + B and the limiting distribution is the unique probability distribution satisfying πQ = 0. Example 2. Let (X t : t ≥ 0) be a time-homogeneous Markov process with the transition rate diagram

For a long time, one of my dreams was to describe the nature of uncertainty axiomatically, and it looks like I've finally done it in my co∼eventum mechanics! Now it remains for me to explain to everyone the co∼eventum mechanics in the most approachable way. This is what I'm trying to do in this work. The co∼eventum mechanics is another name for the co∼event theory, i.e., for the theory of experience and chance which I axiomatized in 2016 [1, 2]. In my opinion, this name best reflects the co∼event-based idea of the new dual theory of uncertainty, which combines the probability theory as a theory of chance, with its dual half, the believability theory as a theory of experience. In addition, I like this new name indicates a direct connection between the co∼event theory and quantum mechanics, which is intended for the physical explanation and description of the conict between quantum observers and quantum observations [4]. Since my theory of uncertainty satises the Kolmogorov axioms of probability theory, to explain this co∼eventum mechanics I will use a way analogous to the already tested one, which explains the theory of probability as a theory of chance describing the results of a random experiment. The simplest example of a random experiment in probability theory is the " tossing a coin ". Therefore, I decided to use this the simplest random experiment itself, as well as the two its analogies: the " "flipping a coin " and the " spinning a coin " to explain the co∼eventum mechanics, which describes the results of a combined experienced random experiment. I would like to resort to the usual for the probability theory " coin-based " analogy to explain (and first of all for myself) the logic of the co∼eventum mechanics as a logic of experience and chance. Of course, this analogy one may seem strange if not crazy. But I did not come up with a better way of tying the explanations of the logic of the co∼eventum mechanics to the coin-based explanations that are commonly used in probability theory to explain at least for myself the logic of the chance through a simple visual " coin-based " model that clarifies what occurs as a result of a combined experienced random experiment in which the experience of observer faces the chance of observation. I hope this analogy can be useful not only for me in understanding the co∼eventum mechanics.

When rolling a single die, the probability of generating any random number is 1/n, were n is the number of faces on the die.  For an icosahedron, numbered 1-20, the probability of generating any particular number is 1/20, or a 5% chance.

For a cube, numbered 1-6, the probability of generating any particular random number is 1/6, or a 16.67% chance.

When rolling multiple dice, the probability of generating any particular sum follows a normal distribution, rather than a linear distribution, with the shape of the bell curve becoming more defined as the number of dice rolls increases.

In this paper, I will explain why the character generation algorithm of Dungeons & Dragons suggests rolling four six-sided dice, and removing the lowest roll, for each attribute.

You yourself, or what is the same, your experience is such ``coin'' that, while you aren't questioned, it rotates all the time in ``free flight''. And only when you answer the question the ``coin'' falls on one of the sides: ``Yes'' or ``No'' with the believability that your experience tells you.

Author of the monograph, Juraj Králik, has been working at the Department of
Structural Mechanics as assistant since September 1, 1976 and as associate professor since
January 18, 1988. During the years 2000 - 2006 he was the head of Department. He holds
lectures in two study programs: Engineering Structures and Transport Structures; and Civil
Engineering and Architecture. He teaches the following subjects: Mechanics of Structures and
Materials, Seismic Enginering and Computer Design, Risk Engineering, Safety and
Reliability of Buildings. His regular students and doctoral students have won several prizes in
the student research competitions at the Faculty. He has been implementing advanced
computer programs and methods in teaching and research at the Department, being himself an
author of more than 100 programs of the static and dynamic applications. He is guarantor of
using the licensed software systems ANSYS and MathCAD at the Department and Faculty,
too. As the co-guarantor, he has established the doctoral study program “Applied Mechanics”.
During the years 2000-2006, in cooperation with the Slovak Chamber of the Civil Engineers
he was the supervisor of seven volumes of the postgraduate study course „Aeroelasticity and
Seismicity“, and the chairman of seven volumes of the International Conference „New Trends
in the Statics and Dynamics of Buildings“.
His results were presented in more than 300 papers in conference proceedings and
journals, 10 papers are indexed in prestigious database „Web of Science“. Three papers were
published in the currented international journals „Mathematics and Computers in Simulation”
(1999), „Control and Cybernetics” (2006) and “Engineering Structures” (2009). 17 research
and grant projects were managed by him. His works were cited in more than 200 papers in
scientific and special publications. As the reputable scientific personality he was the member
of the scientific committees of several international conferences abroad. In the year 1989 he
was appointed an expert of the safety and reliability of nuclear power plants in Slovakia. He
cooperated at the analysis of the seismic resistance of the nuclear power plant buildings and
their safety under impact of explosion, missile and container drop. More than 100 expertises
were realized by him in the field of the safety and reliability of the NPP buildings in Slovakia.
Some of his research and expert works were awarded by significant institutions. The
most significant is the honorable award of the Czech Engineering Academy for the paper
„Probability Analysis of Reinforced Concrete Structure Failure of Nuclear Power Plants Due
to Loss of Coolant Accident “ published in the journal „ENGINEERING MECHANICS“ in
2006.

This half-day tutorial on Belief function (random sets) for the working scientist was presented on July 9th 2016 at the latest International Joint Conference on Artificial Intelligence (IJCAI-16).

The tutorial is very comprehensive (468 slides), covering:

(i) a review of mathematical probability and its interpretations (Bayesian and frequentist);

(ii) the rational for going beyond standard probability: it's all about the data!

(iii) the basis notions of the theory of belief functions;

(iv) reasoning with belief functions: inference, combination/conditioning, graphical models, decision making;

(v) using belief functions for classification, regression, estimation, etc;

(vi) dealing with computational issues and extending belief measures to real numbers;

(vii) the main frameworks derived from belief theory, and its relationship with other theories of uncertainty;

(viii) a number of example applications;

(ix) new horizons, from the formulation of limit theorems for random sets, generalising the notion of likelihood and logistic regression for rare event estimation, climatic change modelling and new foundations for machine learning based on random set theory, a geometry of uncertainty.

Tutorial slides are downloadable at http://cms.brookes.ac.uk/staff/FabioCuzzolin/files/IJCAI2016.pdf

Desde hace décadas, los científicos reclaman insistentemente la necesidad de que los laboratorios forenses abandonen, en la formulación de las conclusiones de los informes periciales, el paradigma de la individualización –que confía en la capacidad de las pruebas científicas para identificar plenamente a un individuo o a un objeto a partir de vestigios– y lo sustituyan por el de la verosimilitud. El presente trabajo pretende ilustrar sobre la trascendencia de este nuevo paradigma y argumentar en favor de este cambio.

The principal aim of this book is to introduce to the widest possible audience an original view of belief calculus and uncertainty theory. In this geometric approach to uncertainty, uncertainty measures can be seen as points of a suitably complex geometric space, and manipulated in that space, for example, combined or conditioned.

In the chapters in Part I, Theories of Uncertainty, the author offers an extensive recapitulation of the state of the art in the mathematics of uncertainty. This part of the book contains the most comprehensive summary to date of the whole of belief theory, with Chap. 4 outlining for the first time, and in a logical order, all the steps of the reasoning chain associated with modelling uncertainty using belief functions, in an attempt to provide a self-contained manual for the working scientist. In addition, the book proposes in Chap. 5 what is possibly the most detailed compendium available of all theories of uncertainty. Part II, The Geometry of Uncertainty, is the core of this book, as it introduces the author’s own geometric approach to uncertainty theory, starting with the geometry of belief functions: Chap. 7 studies the geometry of the space of belief functions, or belief space, both in terms of a simplex and in terms of its recursive bundle structure; Chap. 8 extends the analysis to Dempster’s rule of combination, introducing the notion of a conditional subspace and outlining a simple geometric construction for Dempster’s sum; Chap. 9 delves into the combinatorial properties of plausibility and commonality functions, as equivalent representations of the evidence carried by a belief function; then Chap. 10 starts extending the applicability of the geometric approach to other uncertainty measures, focusing in particular on possibility measures (consonant belief functions) and the related notion of a consistent belief function. The chapters in Part III, Geometric Interplays, are concerned with the interplay of uncertainty measures of different kinds, and the geometry of their relationship, with a particular focus on the approximation problem. Part IV, Geometric Reasoning, examines the application of the geometric approach to the various elements of the reasoning chain illustrated in Chap. 4, in particular conditioning and decision making. Part V concludes the book by outlining a future, complete statistical theory of random sets, future extensions of the geometric approach, and identifying high-impact applications to climate change, machine learning and artificial intelligence.

The book is suitable for researchers in artificial intelligence, statistics, and applied science engaged with theories of uncertainty. The book is supported with the most comprehensive bibliography on belief and uncertainty theory.

This study is motivated by the sharp contrast between physical and probabilistic models of civil engineering. The current practice focuses on physical models while probabilistic ones are relatively underdeveloped. This unbalance can even hinder advances in physical models. Moreover, this rarely acknowledged asymmetry creates the illusion that our deterministic models accurately capture all or at least the main aspects of reality. Thus, this thesis aspires to subtly adjust this imbalance by focusing on probabilistic models.

The main contribution is that it explores often neglected or oversimplified aspects of probabilistic analysis in civil engineering. These distinct but related issues are: (i ) selection of an appropriate distribution type; (ii ) effect of statistical uncertainty; (iii ) measurement uncertainty; (iv) long-term trends; and (v) dependence structure. These are demonstrated through analyzing extreme ground snow loads, although they are inevitably present for most random variables. Snow action, which has recently led to numerous structural failures in Central Europe, is treated as a vehicle of illustration to give a sharp focus to the study.

Methods developed in mathematical statistics, probability theory, information theory, and structural reliability are applied to tackle these issues. Fully statistical analysis of snow extremes is undertaken in conjunction with structural reliability analysis. The popular civil engineering approaches are compared with more advanced techniques that are able to capture the neglected effects in the former approaches. Snow water equivalent data of the Carpathian Region from more than 600 meteorological stations are analyzed, thus the results are representative for lowlands, for highlands, and for mountains as well. Furthermore, extensive parametric analyses are performed to extend the investigations to random variables other than ground snow intensity.

Based on the analysis of the above-listed issues the following main conclusions are drawn:

(i) It is demonstrated that mountains and highlands are better represented by Weibull, while lowlands by Fréchet distribution. In comparison, the currently standardized Gumbel model recommended in Eurocode often appreciably underestimates the snow maxima of lowlands and thus leads to overestimation of structural reliability.

(ii) It is shown that current practice, which neglects statistical uncertainties (parameter estimation and model selection uncertainties), can yield to even multiple orders of magnitude underestimation of failure probability. Bayesian posterior predictive distribution and Bayesian model averaging is proposed to account for parameter estimation and model selection uncertainties, respectively.

(iii) Statistical and interval based approaches are used to explore the effect of prevalently neglected measurement uncertainty on structural reliability. It is demonstrated that such simplification can lead to an order of magnitude underestimation of failure probability. If sufficient data are available to infer the probabilistic model of measurement uncertainty, then the statistical approach is recommended. Otherwise, the interval approach is advocated.

(iv) Using non-stationary extreme value analysis, statistically significant decreasing trend is found in annual ground snow extremes for most of the Carpathian Region. For some locations the effect of the trend on structural reliability is practically significant. This change is favorable from safety point of view as it increases reliability. Hence, revision of current regulations due to long-term trends is not needed from safety reasons, though it might be desirable from economic considerations. However, record lengths are insufficient to draw strong conclusions and to include trends in predicting extreme values with return periods of hundreds of years.

(v) Study of the widely used Gauss (normal, or Gaussian) copula assumption of time-continuous stochastic processes is performed. It reveals that Gauss copula can four times underestimate or even ten times overestimate time-variant failure probabilities obtained by other adopted copulas (t, Gumbel, rotated Gumbel, and rotated Clayton). Model averaging is proposed as a viable approach to rigorously account for copula function uncertainty.

Based on these findings, practical recommendations are made for normal and safety critical structures. The effects and proposed approaches are illustrated through real life examples. Reliability of the more than 130 years old wrought iron structure of Eiffel-hall and a steel hall of Paks Nuclear Power Plant is analyzed. The tackled challenges are general and relevant for most random variables such as wind, traffic, and earthquake actions. Therefore, the findings can be applied for these as well and they can help to draft more consistent standards and to build safer structures.

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

EEC'2016 ~ workshop on axiomatizing experience and chance, and Hilbert's sixth problem

With topics from quantum physics, probability and believability to economics, sociology, and psychology, the workshop will be intended for an interdisciplinary discussion on mathematical theories of experience and chance. Topics of discussion include the results, thoughts, and ideas on the axiomatization of the eventological theory of experience and chance in the framework of the decision of Hilbert sixth problem.
Eventology of experience and chance
Believability theory and statistics of experience
Probability theory and statistics of chance
Axiomatizing experience and chance

Although born within the remit of mathematical statistics, the theory of belief functions has later evolved towards subjective interpretations which have distanced it from its mother field, and have drawn it nearer to artificial intelligence.
The purpose of this talk, in its first part, is to understanding belief theory in the context of mathematical probability and its main interpretations, Bayesian and frequentist statistics, contrasting these three methodologies according to their treatment of uncertain data.
In the second part we recall the existing statistical views of belief function theory, due to the work by Dempster, Almond, Hummel and Landy, Zhang and Liu, Walley and Fine, among others.
Finally, we outline a research programme for the development of a fully-fledged theory of statistical inference with random sets. In particular, we discuss the notion of generalised lower and upper likelihoods, the formulation of a framework for logistic regression with belief functions, the generalisation of the classical total probability theorem to belief functions, the formulation of parametric models based of random sets, and the development of a theory of random variables and processes in which the underlying probability space is replaced by a random set space.

We exist in the world of uncertainties.
Any uncertainty always arises from a conflict of experience and chance, more precisely, from a conflict between the observer's experience and the chance observation. In other words, this indivisible pair, ``experience and chance'', is the source of any uncertainty.
Other sources of uncertainty simply do not exist.
A theory that describes this conflict strictly mathematically is the new theory, called theory of co~events, or the theory of experience and chance. Now we can say that this theory is a theory of uncertainty in its broadest sense.

The eventology approach gave impetus to the development of a theory that turned out to be broader than the theory of probabilities.
This new theory is a dual combination of two theories - Kolmogorov's theory of probabilities of ket-events (k-e.'s) and its dual reflection - a new theory of believabilities of bra-events (b-e.'s). Today, the theory of co~events has a strict axiomatics, in which Kolmogorov k-e.'s describing the future case of observation are dually reflected in b-e.'s describing the past experience of the observer, and a co~event is defined as a measurable binary relation on the Cartesian product: ``a set of bra-events X a set of terraced ket-events''.

The book about the eventology ~ a science about events and its applications to problems of management of sets of events ~ a new direction in philosophy, mathematics and event management (see English version of the book: https://www.academia.edu/38605343/).
This book outlines ideas that inspired me while I was trying to build a new event science, eventology. Not each of these ideas was developed, but all of them inspired me in the construction of eventology, and then in the discovery of a revolutionary theory of experience and chance, co~eventum mechanics. (see, for example, https://www.academia.edu/34357251,
https://www.academia.edu/34373279,
https://www.academia.edu/34390291,
https://www.academia.edu/38368776,
https://www.academia.edu/38154263 and so on).

This work presents results from the investigation of five projects of the Scientific Grant
Agency of Ministry of Education of Slovak Republic (VEGA), three projects of Slovak
Research and the Development Agency of Slovak Republic (APVV), Tempus and Phare
projects during the last 20 years, the more than 100 projects for the Slovak Nuclear
Companies and Nuclear Authority of Slovak Republic and 300 papers in international
conferences and journals. Some most importante are mentioned in [P1-P126].
Overall scientific goal in these projects was complex analysis of existing results from the
point of view of safety and reliability increase of Nuclear Power Plants buildings in Jaslovské
Bohunice and Mochovce and comparative studies of the Slovak standard with international
standard requirements.
The countries with nuclear power plants are unified in the organization OECD, which
organizes forum for publication of experience, development of methodologies and new
methods in the domain of the safety and reliability of nuclear structures and technologies.
The OECD Nuclear Energy Agency (NEA) was established on the 1st February 1958
under the name of the OEEC European Nuclear Energy Agency. It received its present
designation on the 20th April 1972, when Japan became its first non-European full member.
NEA membership today consists of 28 OECD member countries: Australia, Austria, Belgium,
Canada, the Czech Republic, Denmark, Finland, France, Germany, Greece, Hungary, Iceland,
Ireland, Italy, Japan, Luxembourg, Mexico, the Netherlands, Norway, Portugal, the Republic
of Korea, the Slovak Republic, Spain, Sweden, Switzerland, Turkey, the United Kingdom and
the United States. The Commission of the European Communities also takes part in the work
of the Agency.
The primary objective of the NEA is to promote co-operation among governments of its
participating countries in furthering the development of nuclear power as a safe,
environmentally acceptable and economic energy source. This work presents the experiences, methodologies and results from the analysis of the
resistance of nuclear power plant structures to seismic, explosion and impact loads in
accordance to international standards. This work doe not provide complex information of the
safety and reliability problems of NPP because these problems are very large.
There are described new information and new knowledge in the analysis of the safety and
reliability of NPP structures in Slovakia in comparison with international requirements. Two
different systems NPP V1 (Figure 1.2) and NPP V2 (Figure 1.3) were used in Slovakia. The
principal differences between this NPP are in the emergency systems. The NPP V1 was the
800m3 emergency basin for elimination of the accident effects. In the case of NPP V2 the
bubbler tower is used for elimination of accident effects.
The majority of NPP in the word were designed for 30-40 years. In the present the
investigation of nuclear technologies is concentrated to problems of increase of their lifetime
to required level of safety. In the US the lifetime of two NPP was increased to 60 years after
the reconstruction project due to results of experimental and numerical analysis of American
and French companies - Energy and Framatome. The experimental results of the last year on the same nuclear power plant buildings with
reactor WWER 440 in Paks, realized by some independent experts, show more limitations
input suppositions for the numerical seismic analysis and consequently the input parameters
and the methods were modified acceptance soil-structures interaction.
In the last years the risk analysis of the NPP performance is very actual from the point of
view of safety and reliability of the NPP. The risk analysis is based on the probabilistic
analysis method to consider the uncertainties of the input parameters (material, geometry,
loads, model, methods,…). The probabilistic nonlinear analysis of the containment integrity is
presented there.
The results of the presented analysis were used for civil engineering postgraduate training
at Faculty, the seminar in VÚJE Trnava and VÚEZ Levice, which guarantied the quality
program requalification of the existing NPP buildings and technologies in Jaslovské Bohunice
and Mochovce.

A new generalized distribution is developed, namely, Topp-Leone Marshall-Olkin-G distribution. The new distribution is a linear combination of the exponentiated-G family of distributions. We considered three sub-families of the new proposed family of distribution. The distribution can handle heavy-tailed data and various forms of the hazard rate functions. A simulation study was conducted to evaluate consistency of the model parameters. Three applications are provided to demonstrate the usefulness of the new model in comparison with competing non-nested models.

Average case complexity, in order to be a useful and reliable measure, has to be robust. The probability distribution, generally uniform, over which expectation is taken should be realistic over the problem domain. But algorithm books do not certify that the uniform inputs are always realistic. Do the results hold even for non uniform inputs? In this context we observe that Binary Tree sort is more robust than the fast and popular Quick sort in the average case.

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters.

Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.

Practitioners and construction management researchers lack believable and practical methods to assess the value proposition of emerging methods such as Virtual Design and Construction (VDC) including understanding how different levels of implementation affect its benefits. Furthermore, current methods of understanding VDC implementation and benefits cannot be updated easily to incorporate new data. This paper presents a Bayesian framework to predict benefits from application of Virtual Design and Construction (VDC) given data about its implementation. We analyzed data from 40 projects that performed some formal modeling of the project scope and/or the construction process. The analysis suggests that more extensive or higher levels of VDC implementation lead to higher project benefits. We explain the use of a Bayesian framework as an alternative to the application of classical probability theory to construction management research, how we used it to interpret data about VDC practice and outcomes, our finding that benefits have strong positive contingent correlation with the level of VDC implemented on projects, and our suggestion to use the method to update conclusions about benefits given changing data about implementation and outcomes.

Probability theory and statistical evidence can be very helpful to the law; but they can also be misused. Haack explains why by reference to both civil (toxic-tort) and criminal cases.

Presented at the University of Girona, March 2020, on the occasion of Haack's being awarded the Premio internacional de cultura juridica.

The evaluation of fitted binary logistic regression model is very important in assessing the appropriateness of a model for specific purposes. The study proposes to assess the discriminatory performance of a binary logistic regression model to correctly classify between the cases and non-cases. The discriminatory performance of binary logistic regression model is measured using two approaches. The first approach is the use of fitted binary logistic regression model to correctly predict the subjects that are cases and non-cases, with the help of the parameters sensitivity and specificity. The alternative approach is based on receiver operating characteristic (ROC) curve for the fitted binary logistic regression model and then determining the area under the curve (AUC) as a measure of discriminatory performance. The value of sensitivity is observed to be greater than the value of 1-specificity, which signifies suitable discrimination for the mentioned cut point. The area under the curve indicates that there is evidence of reasonable discrimination reported by the fitted model.

For certain algorithms such as sorting and searching, the parameters of the input probability distribution, in addition to the size of the input, have been found to influence the complexity of the underlying algorithm. The present paper makes a statistical comparative study on parameterized complexity between linear and binary search algorithms for binomial inputs.

New in the probability theory and eventology theory, the concept of Kopula (eventological copula) is introduced. The theorem on the characterization of the sets of events by Kopula is proved, which serves as the eventological pre-image of the well-known Sclar's theorem on copulas (1959). The Kopulas of doublets and triplets of events are given, as well as of some N-sets of events. For further development of this topic, see my later works: https://www.academia.edu/34390291/, https://www.academia.edu/34373279/, https://www.academia.edu/34357251/

Keywords: eventology, probability, Kolmogorov event, event, set of events, Kopula (eventological copula), Kopula characterizing a set of events.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and of chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e.g., a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments. Some popular definitions are: Merriam-Webster dictionary defines statistics as "a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data. Statistician Sir Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation. Two main statistical methods are used in data analysis: descriptive statistics, which summarizes data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draws conclusions from data that are subject to random variation (e.g., observational errors, sampling variation).[5] Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.

In this article, we consider a three parameter extended Burr-III distribution and study some distributional, reliability properties and parameter estimation. Performance of estimation technique used for model parameters estimation is numerically investigated employing Monte Carlo simulation with different sample sizes and parameter values. Efficacy of this distribution in modeling from two real life data is evaluated in comparison to some existing extensions of Burr-III distribution employing well known goodness of fit tests and model selection criteria. Our findings show the proposed distribution as the best among the all the other extensions of Burr-III distribution considered in this study. ______________________________

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

EEC'2016 ~ workshop on axiomatizing experience and chance, and Hilbert's sixth problem

With topics from quantum physics, probability and believability to economics, sociology and psychology, the workshop will be intended for an interdisciplinary discussion on mathematical theories of experience and chance. Topics of discussion include the results, thoughts and ideas on axiomatization of the eventological theory of experience and chance in the framework of the decision of Hilbert sixth problem.
Eventology of experience and chance
Beleivability theory and statistics of experience
Probability theory and statistics of chance
Axiomatizing experience and chance

Eventology of multivariate statistics
Eventology and mathematical eventology
Philosophical eventology and philosophy of probability
Practical eventology
Eventology of safety
Eventological economics and psychology
Mathematics in the humanities, socio-economic and natural sciences
Financial and actuarial mathematics
Multivariate statistical analysis
Multivariate complex analysis
Decision-making under risk and uncertainty
Risk measurement and risk models
Theory of fuzzy events and generalized theory of uncertainty
System analysis and events management

This note contains some of the key facts about various common statistical distributions. In the case if continuous distributions, the probability density function p(x) equals the derivative of the cumulative distribution function F (x) = P (X ≤ x). In the case of discrete distributions, the probability density function p(x) equals the probability that the random variable X takes the value of x.

In example-based human pose estimation, the configuration of an evolving object is sought given visual evidence, having to rely uniquely on a set of sample images. We assume here that, at each time instant of a training session, a number of feature measurements is extracted from the available images, while ground truth is provided in the form of the true object pose. In this scenario, a sensible approach consists in learning maps from features to poses, using the information provided by the training set. In particular, multi-valued mappings linking feature values to set of training poses can be constructed. To this purpose we propose a Belief Modeling Regression (BMR) approach in which a probability measure on any individual feature space maps to a convex set of probabilities on the set of training poses, in a form of a belief function. Given a test image, its feature measurements translate into a collection of belief functions on the set of training poses which, when combined, yield there an entire family of probability distributions. From the latter either a single central pose estimate or a set of extremal ones can be computed, together with a measure of how reliable the estimate is. Contrarily to other competing models, in BMR the sparsity of the training samples can be taken into account to model the level of uncertainty associated with these estimates. We illustrate BMR's performance in an application to human pose recovery, showing how it outperforms our implementation of both Relevant Vector Machine and Gaussian Process Regression. Finally, we discuss motivation and advantages of the proposed approach with respect to its most direct competitors. Index Terms Example-based pose estimation, feature-pose maps, theory of evidence, belief functions.

The aim of the paper is the axiomatic justification of the theory of experience and chance, one of the dual halves of which is the Kolmogorov probability theory. The author's main idea was the natural inclusion of Kolmogorov's axiomatics of probability theory in a number of general concepts of the theory of experience and chance. The analogy between the measure of a set and the probability of an event has become clear for a long time. This analogy also allows further evolution: the measure of a set is completely analogous to the believability of an event. In order to postulate the theory of experience and chance on the basis of this analogy, you just need to add to the Kolmogorov probability theory its dual reflection - the believability theory, so that the theory of experience and chance could be postulated as the certainty (believability-probability)  theory on the Cartesian product of the probability and believability spaces, and the central concept of the theory is the new notion of co~event as a measurable binary relation on the Cartesian product of sets of elementary incomes and elementary outcomes. Attempts to build the foundations of the theory of experience and chance from this general point of view are unknown to me, and the whole range of ideas presented here has not yet acquired popularity even in a narrow circle of specialists; in addition, there was still no complete system of the postulates of the theory of experience and chance free from unnecessary complications. Postulating the theory of experience and chance can be carried out in different ways, both in the choice of axioms and in the choice of basic concepts and relations. If one tries to achieve the possible simplicity of both the system of axioms and the theory constructed from it, then it is hardly possible to suggest anything other than axiomatization of concepts co~event and its certainty (believability-probability). The main result of this work is the axiom of co~event, intended for the sake of constructing a theory formed by dual theories of believabilities and probabilities, each of which itself is postulated by its own Kolmogorov system of axioms. Of course, other systems of postulating the theory of experience and chance can be imagined, however, in this work, a preference is given to a system of postulates that is able to describe in the most simple manner the results of what I call an experienced-random experiment.

Keywords: Eventology, event, co~event, experience, chance, to experience, to happen, to occur, theory of experience and chance, theory of co~events, axiom of co~event, probability, believability, certainty (believability-probability), probability theory, believability theory, certainty theory.

If you are interested in my work, and you have the opportunity to provide financial support for the publication of my book "Theory of experience and of chance as a theory of being," I ask you to use my PayPal account: paypal.me/EventologyTheory
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In this article applied and theoretical epistemologies benefit each other in a study of the British legal case of R. vs. Clark. Clark's first infant died at 11 weeks of age, in December 1996. About a year later, Clark had a second child. After that child died at eight weeks of age, Clark was tried for murdering both infants. Statisticians and philosophers have disputed how to apply Bayesian analyses to this case, and thereby arrived at different judgments about it. By dwelling on this applied case, I make theoretical gains: clarifying and defending pragmatic principles of inference that are important for estimating key probabilities in a range of cases. Then, partly by drawing on such principles, and uncovering overlooked data on post-partum psychosis, I make applied gains: improving the rationality of judgments about the Sally Clark case in particular, judgments important to future similar cases.

In this study, we proposed a generalization of the Pranav distribution by Shukla (2018). This new distribution called an extended Pranav distribution is obtained using the exponentiation method. The statistical characteristics of this new distribution such as the moments, moment generating function, reliability function, hazard function, Rényi entropy and order statistics are derived. The graphical illustrations of the shapes of the probability density function, the cumulative distribution function, and hazard rate functions are provided. The maximum likelihood estimates of the parameters were obtained and finally, we examine the performance of this new distribution using some real-life data sets to show its flexibility and better goodness of fit as compared with other distributions.

The main theorem of the games theory of random coalitions is reformulated in the random set language which generalizes the classical maximin theorem but unlike it defines a coalition imputation also. The theorem about maximin random coalitions has been introduced as a ...

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters.

Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.