A1 Probability and Statistics
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
Package 'Prevalence'
Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch Year: 2013 Package ‘prevalence’ Devleesschauwer, Brecht ; Torgerson, Paul R ; Charlier, Johannes ; Levecke, Bruno ; Praet, Nicolas ; Dorny, Pierre ; Berkvens, Dirk ; Speybroeck, Niko Abstract: Tools for prevalence assessment studies. IMPORTANT: the truePrev functions in the preva- lence package call on JAGS (Just Another Gibbs Sampler), which therefore has to be available on the user’s system. JAGS can be downloaded from http://mcmc-jags.sourceforge.net/ Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-89061 Scientific Publication in Electronic Form Published Version The following work is licensed under a Software: GNU General Public License, version 2.0 (GPL-2.0). Originally published at: Devleesschauwer, Brecht; Torgerson, Paul R; Charlier, Johannes; Levecke, Bruno; Praet, Nicolas; Dorny, Pierre; Berkvens, Dirk; Speybroeck, Niko (2013). Package ‘prevalence’. On Line: The Comprehensive R Archive Network. Package ‘prevalence’ September 22, 2013 Type Package Title The prevalence package Version 0.2.0 Date 2013-09-22 Author Brecht Devleesschauwer [aut, cre], Paul Torgerson [aut],Johannes Charlier [aut], Bruno Lev- ecke [aut], Nicolas Praet [aut],Pierre Dorny [aut], Dirk Berkvens [aut], Niko Speybroeck [aut] Maintainer Brecht Devleesschauwer <[email protected]> BugReports https://github.com/brechtdv/prevalence/issues Description Tools for prevalence -
Creating Modern Probability. Its Mathematics, Physics and Philosophy in Historical Perspective
HM 23 REVIEWS 203 The reviewer hopes this book will be widely read and enjoyed, and that it will be followed by other volumes telling even more of the fascinating story of Soviet mathematics. It should also be followed in a few years by an update, so that we can know if this great accumulation of talent will have survived the economic and political crisis that is just now robbing it of many of its most brilliant stars (see the article, ``To guard the future of Soviet mathematics,'' by A. M. Vershik, O. Ya. Viro, and L. A. Bokut' in Vol. 14 (1992) of The Mathematical Intelligencer). Creating Modern Probability. Its Mathematics, Physics and Philosophy in Historical Perspective. By Jan von Plato. Cambridge/New York/Melbourne (Cambridge Univ. Press). 1994. 323 pp. View metadata, citation and similar papers at core.ac.uk brought to you by CORE Reviewed by THOMAS HOCHKIRCHEN* provided by Elsevier - Publisher Connector Fachbereich Mathematik, Bergische UniversitaÈt Wuppertal, 42097 Wuppertal, Germany Aside from the role probabilistic concepts play in modern science, the history of the axiomatic foundation of probability theory is interesting from at least two more points of view. Probability as it is understood nowadays, probability in the sense of Kolmogorov (see [3]), is not easy to grasp, since the de®nition of probability as a normalized measure on a s-algebra of ``events'' is not a very obvious one. Furthermore, the discussion of different concepts of probability might help in under- standing the philosophy and role of ``applied mathematics.'' So the exploration of the creation of axiomatic probability should be interesting not only for historians of science but also for people concerned with didactics of mathematics and for those concerned with philosophical questions. -
There Is No Pure Empirical Reasoning
There Is No Pure Empirical Reasoning 1. Empiricism and the Question of Empirical Reasons Empiricism may be defined as the view there is no a priori justification for any synthetic claim. Critics object that empiricism cannot account for all the kinds of knowledge we seem to possess, such as moral knowledge, metaphysical knowledge, mathematical knowledge, and modal knowledge.1 In some cases, empiricists try to account for these types of knowledge; in other cases, they shrug off the objections, happily concluding, for example, that there is no moral knowledge, or that there is no metaphysical knowledge.2 But empiricism cannot shrug off just any type of knowledge; to be minimally plausible, empiricism must, for example, at least be able to account for paradigm instances of empirical knowledge, including especially scientific knowledge. Empirical knowledge can be divided into three categories: (a) knowledge by direct observation; (b) knowledge that is deductively inferred from observations; and (c) knowledge that is non-deductively inferred from observations, including knowledge arrived at by induction and inference to the best explanation. Category (c) includes all scientific knowledge. This category is of particular import to empiricists, many of whom take scientific knowledge as a sort of paradigm for knowledge in general; indeed, this forms a central source of motivation for empiricism.3 Thus, if there is any kind of knowledge that empiricists need to be able to account for, it is knowledge of type (c). I use the term “empirical reasoning” to refer to the reasoning involved in acquiring this type of knowledge – that is, to any instance of reasoning in which (i) the premises are justified directly by observation, (ii) the reasoning is non- deductive, and (iii) the reasoning provides adequate justification for the conclusion. -
The Interpretation of Probability: Still an Open Issue? 1
philosophies Article The Interpretation of Probability: Still an Open Issue? 1 Maria Carla Galavotti Department of Philosophy and Communication, University of Bologna, Via Zamboni 38, 40126 Bologna, Italy; [email protected] Received: 19 July 2017; Accepted: 19 August 2017; Published: 29 August 2017 Abstract: Probability as understood today, namely as a quantitative notion expressible by means of a function ranging in the interval between 0–1, took shape in the mid-17th century, and presents both a mathematical and a philosophical aspect. Of these two sides, the second is by far the most controversial, and fuels a heated debate, still ongoing. After a short historical sketch of the birth and developments of probability, its major interpretations are outlined, by referring to the work of their most prominent representatives. The final section addresses the question of whether any of such interpretations can presently be considered predominant, which is answered in the negative. Keywords: probability; classical theory; frequentism; logicism; subjectivism; propensity 1. A Long Story Made Short Probability, taken as a quantitative notion whose value ranges in the interval between 0 and 1, emerged around the middle of the 17th century thanks to the work of two leading French mathematicians: Blaise Pascal and Pierre Fermat. According to a well-known anecdote: “a problem about games of chance proposed to an austere Jansenist by a man of the world was the origin of the calculus of probabilities”2. The ‘man of the world’ was the French gentleman Chevalier de Méré, a conspicuous figure at the court of Louis XIV, who asked Pascal—the ‘austere Jansenist’—the solution to some questions regarding gambling, such as how many dice tosses are needed to have a fair chance to obtain a double-six, or how the players should divide the stakes if a game is interrupted. -
The Probability Set-Up.Pdf
CHAPTER 2 The probability set-up 2.1. Basic theory of probability We will have a sample space, denoted by S (sometimes Ω) that consists of all possible outcomes. For example, if we roll two dice, the sample space would be all possible pairs made up of the numbers one through six. An event is a subset of S. Another example is to toss a coin 2 times, and let S = fHH;HT;TH;TT g; or to let S be the possible orders in which 5 horses nish in a horse race; or S the possible prices of some stock at closing time today; or S = [0; 1); the age at which someone dies; or S the points in a circle, the possible places a dart can hit. We should also keep in mind that the same setting can be described using dierent sample set. For example, in two solutions in Example 1.30 we used two dierent sample sets. 2.1.1. Sets. We start by describing elementary operations on sets. By a set we mean a collection of distinct objects called elements of the set, and we consider a set as an object in its own right. Set operations Suppose S is a set. We say that A ⊂ S, that is, A is a subset of S if every element in A is contained in S; A [ B is the union of sets A ⊂ S and B ⊂ S and denotes the points of S that are in A or B or both; A \ B is the intersection of sets A ⊂ S and B ⊂ S and is the set of points that are in both A and B; ; denotes the empty set; Ac is the complement of A, that is, the points in S that are not in A. -
Suitability of Different Probability Distributions for Performing Schedule Risk Simulations in Project Management
2016 Proceedings of PICMET '16: Technology Management for Social Innovation Suitability of Different Probability Distributions for Performing Schedule Risk Simulations in Project Management J. Krige Visser Department of Engineering and Technology Management, University of Pretoria, Pretoria, South Africa Abstract--Project managers are often confronted with the The Project Evaluation and Review Technique (PERT), in question on what is the probability of finishing a project within conjunction with the Critical Path Method (CPM), were budget or finishing a project on time. One method or tool that is developed in the 1950’s to address the uncertainty in project useful in answering these questions at various stages of a project duration for complex projects [11], [13]. The expected value is to develop a Monte Carlo simulation for the cost or duration or mean value for each activity of the project network was of the project and to update and repeat the simulations with actual data as the project progresses. The PERT method became calculated by applying the beta distribution and three popular in the 1950’s to express the uncertainty in the duration estimates for the duration of the activity. The total project of activities. Many other distributions are available for use in duration was determined by adding all the duration values of cost or schedule simulations. the activities on the critical path. The Monte Carlo simulation This paper discusses the results of a project to investigate the (MCS) provides a distribution for the total project duration output of schedule simulations when different distributions, e.g. and is therefore more useful as a method or tool for decision triangular, normal, lognormal or betaPert, are used to express making. -
1 Stochastic Processes and Their Classification
1 1 STOCHASTIC PROCESSES AND THEIR CLASSIFICATION 1.1 DEFINITION AND EXAMPLES Definition 1. Stochastic process or random process is a collection of random variables ordered by an index set. ☛ Example 1. Random variables X0;X1;X2;::: form a stochastic process ordered by the discrete index set f0; 1; 2;::: g: Notation: fXn : n = 0; 1; 2;::: g: ☛ Example 2. Stochastic process fYt : t ¸ 0g: with continuous index set ft : t ¸ 0g: The indices n and t are often referred to as "time", so that Xn is a descrete-time process and Yt is a continuous-time process. Convention: the index set of a stochastic process is always infinite. The range (possible values) of the random variables in a stochastic process is called the state space of the process. We consider both discrete-state and continuous-state processes. Further examples: ☛ Example 3. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f0; 1; 2; 3; 4g representing which of four types of transactions a person submits to an on-line data- base service, and time n corresponds to the number of transactions submitted. ☛ Example 4. fXn : n = 0; 1; 2;::: g; where the state space of Xn is f1; 2g re- presenting whether an electronic component is acceptable or defective, and time n corresponds to the number of components produced. ☛ Example 5. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2;::: g representing the number of accidents that have occurred at an intersection, and time t corresponds to weeks. ☛ Example 6. fYt : t ¸ 0g; where the state space of Yt is f0; 1; 2; : : : ; sg representing the number of copies of a software product in inventory, and time t corresponds to days. -
Determinism, Indeterminism and the Statistical Postulate
Tipicality, Explanation, The Statistical Postulate, and GRW Valia Allori Northern Illinois University [email protected] www.valiaallori.com Rutgers, October 24-26, 2019 1 Overview Context: Explanation of the macroscopic laws of thermodynamics in the Boltzmannian approach Among the ingredients: the statistical postulate (connected with the notion of probability) In this presentation: typicality Def: P is a typical property of X-type object/phenomena iff the vast majority of objects/phenomena of type X possesses P Part I: typicality is sufficient to explain macroscopic laws – explanatory schema based on typicality: you explain P if you explain that P is typical Part II: the statistical postulate as derivable from the dynamics Part III: if so, no preference for indeterministic theories in the quantum domain 2 Summary of Boltzmann- 1 Aim: ‘derive’ macroscopic laws of thermodynamics in terms of the microscopic Newtonian dynamics Problems: Technical ones: There are to many particles to do exact calculations Solution: statistical methods - if 푁 is big enough, one can use suitable mathematical technique to obtain information of macro systems even without having the exact solution Conceptual ones: Macro processes are irreversible, while micro processes are not Boltzmann 3 Summary of Boltzmann- 2 Postulate 1: the microscopic dynamics microstate 푋 = 푟1, … . 푟푁, 푣1, … . 푣푁 in phase space Partition of phase space into Macrostates Macrostate 푀(푋): set of macroscopically indistinguishable microstates Macroscopic view Many 푋 for a given 푀 given 푀, which 푋 is unknown Macro Properties (e.g. temperature): they slowly vary on the Macro scale There is a particular Macrostate which is incredibly bigger than the others There are many ways more to have, for instance, uniform temperature than not equilibrium (Macro)state 4 Summary of Boltzmann- 3 Entropy: (def) proportional to the size of the Macrostate in phase space (i.e. -
Final Paper (PDF)
Analytic Method for Probabilistic Cost and Schedule Risk Analysis Final Report 5 April2013 PREPARED FOR: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION (NASA) OFFICE OF PROGRAM ANALYSIS AND EVALUATION (PA&E) COST ANALYSIS DIVISION (CAD) Felecia L. London Contracting Officer NASA GODDARD SPACE FLIGHT CENTER, PROCUREMENT OPERATIONS DIVISION OFFICE FOR HEADQUARTERS PROCUREMENT, 210.H Phone: 301-286-6693 Fax:301-286-1746 e-mail: [email protected] Contract Number: NNHl OPR24Z Order Number: NNH12PV48D PREPARED BY: RAYMOND P. COVERT, COVARUS, LLC UNDER SUBCONTRACT TO GALORATHINCORPORATED ~ SEER. br G A L 0 R A T H [This Page Intentionally Left Blank] ii TABLE OF CONTENTS 1 Executive Summacy.................................................................................................. 11 2 In.troduction .............................................................................................................. 12 2.1 Probabilistic Nature of Estimates .................................................................................... 12 2.2 Uncertainty and Risk ....................................................................................................... 12 2.2.1 Probability Density and Probability Mass ................................................................ 12 2.2.2 Cumulative Probability ............................................................................................. 13 2.2.3 Definition ofRisk ..................................................................................................... 14 2.3 -
Topic 1: Basic Probability Definition of Sets
Topic 1: Basic probability ² Review of sets ² Sample space and probability measure ² Probability axioms ² Basic probability laws ² Conditional probability ² Bayes' rules ² Independence ² Counting ES150 { Harvard SEAS 1 De¯nition of Sets ² A set S is a collection of objects, which are the elements of the set. { The number of elements in a set S can be ¯nite S = fx1; x2; : : : ; xng or in¯nite but countable S = fx1; x2; : : :g or uncountably in¯nite. { S can also contain elements with a certain property S = fx j x satis¯es P g ² S is a subset of T if every element of S also belongs to T S ½ T or T S If S ½ T and T ½ S then S = T . ² The universal set is the set of all objects within a context. We then consider all sets S ½ . ES150 { Harvard SEAS 2 Set Operations and Properties ² Set operations { Complement Ac: set of all elements not in A { Union A \ B: set of all elements in A or B or both { Intersection A [ B: set of all elements common in both A and B { Di®erence A ¡ B: set containing all elements in A but not in B. ² Properties of set operations { Commutative: A \ B = B \ A and A [ B = B [ A. (But A ¡ B 6= B ¡ A). { Associative: (A \ B) \ C = A \ (B \ C) = A \ B \ C. (also for [) { Distributive: A \ (B [ C) = (A \ B) [ (A \ C) A [ (B \ C) = (A [ B) \ (A [ C) { DeMorgan's laws: (A \ B)c = Ac [ Bc (A [ B)c = Ac \ Bc ES150 { Harvard SEAS 3 Elements of probability theory A probabilistic model includes ² The sample space of an experiment { set of all possible outcomes { ¯nite or in¯nite { discrete or continuous { possibly multi-dimensional ² An event A is a set of outcomes { a subset of the sample space, A ½ . -
Location-Scale Distributions
Location–Scale Distributions Linear Estimation and Probability Plotting Using MATLAB Horst Rinne Copyright: Prof. em. Dr. Horst Rinne Department of Economics and Management Science Justus–Liebig–University, Giessen, Germany Contents Preface VII List of Figures IX List of Tables XII 1 The family of location–scale distributions 1 1.1 Properties of location–scale distributions . 1 1.2 Genuine location–scale distributions — A short listing . 5 1.3 Distributions transformable to location–scale type . 11 2 Order statistics 18 2.1 Distributional concepts . 18 2.2 Moments of order statistics . 21 2.2.1 Definitions and basic formulas . 21 2.2.2 Identities, recurrence relations and approximations . 26 2.3 Functions of order statistics . 32 3 Statistical graphics 36 3.1 Some historical remarks . 36 3.2 The role of graphical methods in statistics . 38 3.2.1 Graphical versus numerical techniques . 38 3.2.2 Manipulation with graphs and graphical perception . 39 3.2.3 Graphical displays in statistics . 41 3.3 Distribution assessment by graphs . 43 3.3.1 PP–plots and QQ–plots . 43 3.3.2 Probability paper and plotting positions . 47 3.3.3 Hazard plot . 54 3.3.4 TTT–plot . 56 4 Linear estimation — Theory and methods 59 4.1 Types of sampling data . 59 IV Contents 4.2 Estimators based on moments of order statistics . 63 4.2.1 GLS estimators . 64 4.2.1.1 GLS for a general location–scale distribution . 65 4.2.1.2 GLS for a symmetric location–scale distribution . 71 4.2.1.3 GLS and censored samples . -
Section III Non-Parametric Distribution Functions
DRAFT 2-6-98 DO NOT CITE OR QUOTE LIST OF ATTACHMENTS ATTACHMENT 1: Glossary ...................................................... A1-2 ATTACHMENT 2: Probabilistic Risk Assessments and Monte-Carlo Methods: A Brief Introduction ...................................................................... A2-2 Tiered Approach to Risk Assessment .......................................... A2-3 The Origin of Monte-Carlo Techniques ........................................ A2-3 What is Monte-Carlo Analysis? .............................................. A2-4 Random Nature of the Monte Carlo Analysis .................................... A2-6 For More Information ..................................................... A2-6 ATTACHMENT 3: Distribution Selection ............................................ A3-1 Section I Introduction .................................................... A3-2 Monte-Carlo Modeling Options ........................................ A3-2 Organization of Document ........................................... A3-3 Section II Parametric Methods .............................................. A3-5 Activity I ! Selecting Candidate Distributions ............................ A3-5 Make Use of Prior Knowledge .................................. A3-5 Explore the Data ............................................ A3-7 Summary Statistics .................................... A3-7 Graphical Data Analysis. ................................ A3-9 Formal Tests for Normality and Lognormality ............... A3-10 Activity II ! Estimation of Parameters