Probability Theory

Under the common viewpoint of statistical maps,the concept of observables in quantum mechanics and inclassical probability theory are discussed and compared.It is shown that, by means of injective statistical maps, quantum mechanics can to a certain extentbe reformulated in classical terms. Some characteristicexamples are considered.

In the UK, hospitals are being encouraged to introduce booked admissions policies for elective inpatient services whereby patients are given a date for hospital admission months in advance rather than being put on a waiting list and then informed of their admission date at short notice. We address the question of what level of capacity is required to operate such a system if cancellations of booked elective patients are to be kept to a low level. Methods are presented for quantifying the day to day variation in bed demand due to emergency admissions, patient initiated cancellations and variable lengths of stay amongst patients.

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman–Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.

We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and zero- and unit-probability events pose no particular epistemological problems. We use a non-Archimedean field as the range of the probability function. As a

Distribusi Multivariate Normal (Gaussian) adalah salah satu distribusi yang sering
digunakan, mengingat hampir semua kejadian bisa didekati dengan distribusi tersebut. Dalam
mengenali suatu objek dalam bentuk image, fitur objek tersebut kerapkali mengikuti distribusi
  Multivariate Gaussian dengan parameter mean dan covariance yang berbebeda-beda.
  Parameter dan yang berbeda-beda tersebut akan menghasilkan nilai probability density
function (pdf) yang berbeda pula. Berdasarakan nilai probability density function ini
selanjutnya dapat dibentuk fungsi diskriminan untuk mengenali objek (Gaussian Classifier).
Kehandalan Gaussian Classifer dalam mengenali objek dalam bentuk image dipengaruhi oleh
2 faktor utama yaitu ketepatan dan keakuratan dalam pengambilan data objek training yang
akan berpengaruh terhadap ketepatan dan keakuratan fitur yang diambil dan asumsi distribusi
Multivariate Normal dari fitur objek yang diambil harus terpenuhi. Untuk memenuhi asumsi
multivariate distribusi Multivariate Normal maka harus dilakukan pengujian terhadap
normalitas distribusi fitur setiap kelas objek.

Statistical learning theory explores ways of estimating functional dependency from a given collection of data. The specific sub-area of supervised statistical learning covers important models like Perceptron, Support Vector Machines (SVM) and Li-near Discriminant Analysis (LDA). In this paper we review the theory of such models and compare their separating hypersurfaces for extracting group-differences between samples. Classification and reconstruction are the main goals of this comparison. We show recent advances in this topic of research illustrating their application on face and medical image databases.

The theory of belief functions, sometimes referred to as evidence theory or Dempster-Shafer theory, was first introduced by Arthur P. Dempster in the context of statistical inference, to be later developed by Glenn Shafer as a general framework for modelling epistemic uncertainty. Belief theory and the closely related random set theory form natural frameworks for modelling situations in which data are missing or scarce: think of extremely rare events such as volcanic eruptions or power plant meltdowns, problems subject to huge uncertainties due to the number and complexity of the factors involved (e.g. climate change), but also the all-important issue with generalisation from small training sets in machine learning. This tutorial is designed to introduce the principles and rationale of random sets and belief function theory to mainstream statisticians, mathematicians and working scientists, survey the key elements of the methodology and the most recent developments, make practitioners aware of the set of tools that have been developed for reasoning in the belief function framework on real-world problems. Attendees will acquire first-hand knowledge of how to apply these tools to significant problems in major application fields such as computer vision, climate change, and others. A research programme for the future of random set theory and high impact applications is eventually outlined.

Tversky and Kahneman (1983) showed that when subjects are asked to rate the likelihood of several alternatives, including single and joint events, they often make a “conjunction fallacy.” That is, they rate the conjunction of two events as being more likely than one of the constituent events. This, they claim, is a fallacy, since the conjunction of two events can never be more probable than either of the component events. In addition, they found that prior training in probability theory does not decrease the likelihood of making this fallacy. We argue that in some contexts, an alternative that contains the conjunction oftwo events can be more probable than an alternative that contains only one of the conjunction's constituent events. We carried out four experiments in which we manipulated this context. The frequency of making a conjunction fallacy was affected by the manipulation of context. Furthermore, when the context was clearly specified, prior training in statistics influenced the ratings.

8. HANDBOOK OF MONETARY ECONOMICS (in 2 volumes) Edited by Benjamin Friedman and Frank Hahn 9. HANDBOOK OF DEVELOPMENT ECONOMICS (in 4 volumes) Volumes 1 and 2 edited by Hollis B. Chenery and TN Srinivasan Volumes 3A and 3B edited by Jere ...

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.

Two approaches for constructing control charts to monitor multivariate attribute processes when data is presented in linguistic form are suggested. Two monitoring statistics T \(^{\rm 2}_{f}\) and W 2 are developed based on fuzzy and probability theories. The first is similar to the Hotteling’s T 2 statistic and is based on representative values of fuzzy sets. The W 2 statistic, being a linear combination of dependent chi-square variables, its distribution is derived by Satterthwaite’s approximation. Resulting multivariate control charts are compared based on the average run length (ARL). A numerical example is given to illustrate the application of the proposed multivariate control charts and the interpretation of out-of-control signals.

The relation of the Polya-Aeppli distribution of probabilities (also known as the Poisson-Geometric distribution) with random processes in the stationary case is considered in this article. Then the above mentioned distribution is derived, working with the probability generating functions, as limit of the distribution of rare events in a succession of Bernouilli trials with first order Markov dependence (that is, as a limit of Markov chains of rare events).

We describe a model for strings of characters that is loosely based on the Lempel Ziv model with the addition that a repeated substring can be an approximate match to the original substring; this is close to the situation of DNA, for example. Typically there are many explanations for a given string under the model, some optimal and many suboptimal. Rather than commit to one optimal explanation, we sum the probabilities over all explanations under the model because this gives the probability of the data under the model. The model has a small number of parameters and these can be estimated from the given string by an expectation-maximization (EM) algorithm. Each iteration of the EM algorithm takes O(n2) time and a few iterations are typically sufficient. O(n2) complexity is impractical for strings of more than a few tens of thousands of characters and a faster approximation algorithm is also given. The model is further extended to include approximate reverse complementary repeats when...

On the question of whether gambling behavior can be changed as result of teaching gamblers the mathematics of gambling, past studies have yielded contradictory results, and a clear conclusion has not yet been drawn. In this paper, I bring some criticisms to the empirical studies that tended to answer no to this hypothesis, regarding the sampling and laboratory testing, and I argue that an optimal mathematical scholastic intervention with the objective of preventing problem gambling is possible, by providing the principles that would optimize the structure and content of the teaching module. Given the ethical aspects of the exposure of mathematical facts behind games of chance, and starting from the slots case – where the parametric design is missing, we have to draw a line between ethical and optional information with respect to the mathematical content provided by a scholastic intervention. Arguing for the role of mathematics in problem-gambling prevention and treatment, interdisciplinary research directions are drawn toward implementing an optimal mathematical module in cognitive therapies.

Keywords: gambling mathematics; slot machines; probability theory; interpretations of probability; psychology of probability; gambling ethics.