Stat 451 Lecture Notes 0512 Simulating Random Variables
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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. -
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 -
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 -
Fixed-K Asymptotic Inference About Tail Properties
Journal of the American Statistical Association ISSN: 0162-1459 (Print) 1537-274X (Online) Journal homepage: http://www.tandfonline.com/loi/uasa20 Fixed-k Asymptotic Inference About Tail Properties Ulrich K. Müller & Yulong Wang To cite this article: Ulrich K. Müller & Yulong Wang (2017) Fixed-k Asymptotic Inference About Tail Properties, Journal of the American Statistical Association, 112:519, 1334-1343, DOI: 10.1080/01621459.2016.1215990 To link to this article: https://doi.org/10.1080/01621459.2016.1215990 Accepted author version posted online: 12 Aug 2016. Published online: 13 Jun 2017. Submit your article to this journal Article views: 206 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=uasa20 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION , VOL. , NO. , –, Theory and Methods https://doi.org/./.. Fixed-k Asymptotic Inference About Tail Properties Ulrich K. Müller and Yulong Wang Department of Economics, Princeton University, Princeton, NJ ABSTRACT ARTICLE HISTORY We consider inference about tail properties of a distribution from an iid sample, based on extreme value the- Received February ory. All of the numerous previous suggestions rely on asymptotics where eventually, an infinite number of Accepted July observations from the tail behave as predicted by extreme value theory, enabling the consistent estimation KEYWORDS of the key tail index, and the construction of confidence intervals using the delta method or other classic Extreme quantiles; tail approaches. In small samples, however, extreme value theory might well provide good approximations for conditional expectations; only a relatively small number of tail observations. -
Markov Chain Monte Carlo 1 Importance Sampling
CS731 Spring 2011 Advanced Artificial Intelligence Markov Chain Monte Carlo Lecturer: Xiaojin Zhu [email protected] A fundamental problem in machine learning is to generate samples from a distribution: x ∼ p(x). (1) This problem has many important applications. For example, one can approximate the expectation of a function φ(x) Z µ ≡ Ep[φ(x)] = φ(x)p(x)dx (2) by the sample average over x1 ... xn ∼ p(x): n 1 X µˆ ≡ φ(x ) (3) n i i=1 This is known as the Monte Carlo method. A concrete application is in Bayesian predictive modeling where R p(x | θ)p(θ | Data)dθ. Clearly,µ ˆ is an unbiased estimator of µ. Furthermore, the variance ofµ ˆ decreases as V(φ)/n (4) R 2 where V(φ) = (φ(x) − µ) p(x)dx. Thus Monte Carlo methods depend on the sample size n, not on the dimensionality of x. 1 Often, p is only known up to a constant. That is, we assume p(x) = Z p˜(x) and we are able to evaluate p˜(x) at any point x. However, we cannot compute the normalization constant Z. For instance, in the 1 Bayesian modeling example, the posterior distribution is p(θ | Data) = Z p(θ)p(Data | θ). This makes sampling harder. All methods below work onp ˜. 1 Importance Sampling Unlike other methods discussed later, importance sampling is not a method to sample from p. Rather, it is a method to compute (3). Assume we know the unnormalizedp ˜. It turns out that we (or Matlab) only know how to directly sample from very few distributions such as uniform, Gaussian, etc.; p may not be one of them in general. -
Hand-Book on STATISTICAL DISTRIBUTIONS for Experimentalists
Internal Report SUF–PFY/96–01 Stockholm, 11 December 1996 1st revision, 31 October 1998 last modification 10 September 2007 Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists by Christian Walck Particle Physics Group Fysikum University of Stockholm (e-mail: [email protected]) Contents 1 Introduction 1 1.1 Random Number Generation .............................. 1 2 Probability Density Functions 3 2.1 Introduction ........................................ 3 2.2 Moments ......................................... 3 2.2.1 Errors of Moments ................................ 4 2.3 Characteristic Function ................................. 4 2.4 Probability Generating Function ............................ 5 2.5 Cumulants ......................................... 6 2.6 Random Number Generation .............................. 7 2.6.1 Cumulative Technique .............................. 7 2.6.2 Accept-Reject technique ............................. 7 2.6.3 Composition Techniques ............................. 8 2.7 Multivariate Distributions ................................ 9 2.7.1 Multivariate Moments .............................. 9 2.7.2 Errors of Bivariate Moments .......................... 9 2.7.3 Joint Characteristic Function .......................... 10 2.7.4 Random Number Generation .......................... 11 3 Bernoulli Distribution 12 3.1 Introduction ........................................ 12 3.2 Relation to Other Distributions ............................. 12 4 Beta distribution 13 4.1 Introduction ....................................... -
Handbook on Probability Distributions
R powered R-forge project Handbook on probability distributions R-forge distributions Core Team University Year 2009-2010 LATEXpowered Mac OS' TeXShop edited Contents Introduction 4 I Discrete distributions 6 1 Classic discrete distribution 7 2 Not so-common discrete distribution 27 II Continuous distributions 34 3 Finite support distribution 35 4 The Gaussian family 47 5 Exponential distribution and its extensions 56 6 Chi-squared's ditribution and related extensions 75 7 Student and related distributions 84 8 Pareto family 88 9 Logistic distribution and related extensions 108 10 Extrem Value Theory distributions 111 3 4 CONTENTS III Multivariate and generalized distributions 116 11 Generalization of common distributions 117 12 Multivariate distributions 133 13 Misc 135 Conclusion 137 Bibliography 137 A Mathematical tools 141 Introduction This guide is intended to provide a quite exhaustive (at least as I can) view on probability distri- butions. It is constructed in chapters of distribution family with a section for each distribution. Each section focuses on the tryptic: definition - estimation - application. Ultimate bibles for probability distributions are Wimmer & Altmann (1999) which lists 750 univariate discrete distributions and Johnson et al. (1994) which details continuous distributions. In the appendix, we recall the basics of probability distributions as well as \common" mathe- matical functions, cf. section A.2. And for all distribution, we use the following notations • X a random variable following a given distribution, • x a realization of this random variable, • f the density function (if it exists), • F the (cumulative) distribution function, • P (X = k) the mass probability function in k, • M the moment generating function (if it exists), • G the probability generating function (if it exists), • φ the characteristic function (if it exists), Finally all graphics are done the open source statistical software R and its numerous packages available on the Comprehensive R Archive Network (CRAN∗). -
Field Guide to Continuous Probability Distributions
Field Guide to Continuous Probability Distributions Gavin E. Crooks v 1.0.0 2019 G. E. Crooks – Field Guide to Probability Distributions v 1.0.0 Copyright © 2010-2019 Gavin E. Crooks ISBN: 978-1-7339381-0-5 http://threeplusone.com/fieldguide Berkeley Institute for Theoretical Sciences (BITS) typeset on 2019-04-10 with XeTeX version 0.99999 fonts: Trump Mediaeval (text), Euler (math) 271828182845904 2 G. E. Crooks – Field Guide to Probability Distributions Preface: The search for GUD A common problem is that of describing the probability distribution of a single, continuous variable. A few distributions, such as the normal and exponential, were discovered in the 1800’s or earlier. But about a century ago the great statistician, Karl Pearson, realized that the known probabil- ity distributions were not sufficient to handle all of the phenomena then under investigation, and set out to create new distributions with useful properties. During the 20th century this process continued with abandon and a vast menagerie of distinct mathematical forms were discovered and invented, investigated, analyzed, rediscovered and renamed, all for the purpose of de- scribing the probability of some interesting variable. There are hundreds of named distributions and synonyms in current usage. The apparent diver- sity is unending and disorienting. Fortunately, the situation is less confused than it might at first appear. Most common, continuous, univariate, unimodal distributions can be orga- nized into a small number of distinct families, which are all special cases of a single Grand Unified Distribution. This compendium details these hun- dred or so simple distributions, their properties and their interrelations. -
The Family of Curvi-Triangular Distributions 1 Introduction
International Journal of Statistical Sciences ISSN 1683–5603 Vol. 6 (Special Issue), 2007, pp 7-18 c 2007 Dept. of Statistics, Univ. of Rajshahi, Bangladesh The Family of Curvi-Triangular Distributions Abdus S. Wahed Dept. of Biostatistics Graduate School of Public Health University of Pittsburgh Pittsburgh, PA 15261, USA. [Received September 29, 2005; Revised July 3, 2007; Accepted September 5, 2007] Abstract A family of distributions has been proposed which includes the symmetric triangular distribution as special case. The new family of distributions, which we refer to as generalized curvi-triangular (GCT) distributions, al- lows the shape of the density to be non-triangular and asymmetric, in general. The properties of GCT distributions along with the inferential procedures regarding the parameters are discussed. Keywords and Phrases: Skewness; Skew-Normal distribution; Triangu- lar distribution. AMS Classification: 60Exx. 1 Introduction A regular three-parameter triangular distribution is given by the density 2(x−α) , α x ξ f(x; α,β,ξ)= (β−α)(ξ−α) ≤ ≤ . (1) 2(β−x) , ξ x β (β−α)(β−ξ) ≤ ≤ This distribution has limited application, mostly because of its triangle-shape density, which makes it less suitable for real-life data. In practice, frequency curves seldom take the shape of a triangle. This kept the use of triangular density clinical. We will mention several recent works with triangular distribution here. An extension of the three-parameter triangular distribution utilized in risk analysis has been discussed by Ren´e van Dorp and Kotz (2002). They showed that the extended four-parameter 8 International Journal of Statistical Sciences, Vol. -
Slice Sampling for General Completely Random Measures
Slice Sampling for General Completely Random Measures Peiyuan Zhu Alexandre Bouchard-Cotˆ e´ Trevor Campbell Department of Statistics University of British Columbia Vancouver, BC V6T 1Z4 Abstract such model can select each category zero or once, the latent categories are called features [1], whereas if each Completely random measures provide a princi- category can be selected with multiplicities, the latent pled approach to creating flexible unsupervised categories are called traits [2]. models, where the number of latent features is Consider for example the problem of modelling movie infinite and the number of features that influ- ratings for a set of users. As a first rough approximation, ence the data grows with the size of the data an analyst may entertain a clustering over the movies set. Due to the infinity the latent features, poste- and hope to automatically infer movie genres. Cluster- rior inference requires either marginalization— ing in this context is limited; users may like or dislike resulting in dependence structures that prevent movies based on many overlapping factors such as genre, efficient computation via parallelization and actor and score preferences. Feature models, in contrast, conjugacy—or finite truncation, which arbi- support inference of these overlapping movie attributes. trarily limits the flexibility of the model. In this paper we present a novel Markov chain As the amount of data increases, one may hope to cap- Monte Carlo algorithm for posterior inference ture increasingly sophisticated patterns. We therefore that adaptively sets the truncation level using want the model to increase its complexity accordingly. auxiliary slice variables, enabling efficient, par- In our movie example, this means uncovering more and allelized computation without sacrificing flexi- more diverse user preference patterns and movie attributes bility. -
Rcppdist Introduction JB Duck-Mayr 2018-10-21
RcppDist Introduction JB Duck-Mayr 2018-10-21 The Rcpp package provides a C++ library to make it easier to use C++ with R. R and Rcpp provide functions for a variety of statistical distributions. Several R packages make functions available to R for additional statistical distributions. However, to access these functions from C++ code, a costly call to the R functions must be made.1 RcppDist provides a C++ header-only library with functions for additional statistical distributions that can be called from C++ when writing code using Rcpp or RcppArmadillo. Functions are available that return NumericVectors as well as doubles, and for multivariate or matrix distributions, Armadillo vectors and matrices. RcppDist provides functions for the following distributions: • the four parameter beta distribution • the location-scale t distribution • the truncated normal distribution • the truncated t distribution • a truncated location-scale t distribution • the triangle distribution • the multivariate normal distribution* • the multivariate t distribution* • the Wishart distribution* • the inverse Wishart distribution* Distributions marked with an asterisk rely on RcppArmadillo. After discussing how to include RcppDist headers into your code to use these functions, each function provided by the package is listed (such that you can see the function and argument names, as well as the return and argument types), with a brief explanation. Including RccpDist in Your Code RcppDist provides several header files you can include in your code. RcppDist.h includes all of them, as well as Rcpp.h or RcppArmadillo.h as appropriate (see “Use Rpp, or RcppArmadillo?” below). So, in any C++ file you need to use functions from RcppDist, you can simply use #include <RcppDist.h> which will also take care of Rcpp(Armadillo) headers for you.