Ebookdistributions.Pdf

Total Page:16

File Type:pdf, Size:1020Kb

Ebookdistributions.Pdf DOWNLOAD YOUR FREE MODELRISK TRIAL Adapted from Risk Analysis: a quantitative guide by David Vose. Published by John Wiley and Sons (2008). All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. If you notice any errors or omissions, please contact [email protected] Referencing this document Please use the following reference, following the Harvard system of referencing: Van Hauwermeiren M, Vose D and Vanden Bossche S (2012). A Compendium of Distributions (second edition). [ebook]. Vose Software, Ghent, Belgium. Available from www.vosesoftware.com . Accessed dd/mm/yy. © Vose Software BVBA (2012) www.vosesoftware.com Updated 17 January, 2012. Page 2 Table of Contents Introduction .................................................................................................................... 7 DISCRETE AND CONTINUOUS DISTRIBUTIONS.......................................................................... 7 Discrete Distributions .............................................................................................. 7 Continuous Distributions ......................................................................................... 8 BOUNDED AND UNBOUNDED DISTRIBUTIONS ......................................................................... 9 PARAMETRIC AND NON-PARAMETRIC DISTRIBUTIONS .......................................................... 10 Univariate and multivariate distributions............................................................................... 10 Lists of applications and the most useful distributions ........................................................... 11 Bounded versus unbounded .................................................................................. 11 Frequency distributions ......................................................................................... 13 Risk impact ............................................................................................................ 13 Time or trials until … .............................................................................................. 13 Variations in a financial market .............................................................................. 14 How big something is ............................................................................................. 15 Expert estimates .................................................................................................... 16 How to read probability distribution equations ............................................................. 17 Location, scale and shape parameters ................................................................................... 17 Understanding distribution equations ................................................................................... 18 Probability mass function (pmf) and probability density function (pdf) .................. 18 Cumulative distribution function (cdf) ................................................................... 20 Mean µ .................................................................................................................. 20 Mode ..................................................................................................................... 21 Variance V ............................................................................................................. 21 Skewness S ............................................................................................................ 22 Kurtosis K............................................................................................................... 22 Univariate Distributions ................................................................................................. 24 Bernoulli ............................................................................................................................... 25 Beta ...................................................................................................................................... 27 Beta (4 parameter) ................................................................................................................ 29 Beta-Binomial ........................................................................................................................ 31 Beta-Geometric ..................................................................................................................... 33 Beta-Negative Bininomial ...................................................................................................... 35 Binomial ................................................................................................................................ 37 © Vose Software BVBA (2012) www.vosesoftware.com Updated 17 January, 2012. Page 3 Bradford ................................................................................................................................ 39 Burnt Finger Poisson ............................................................................................................. 41 Burr ....................................................................................................................................... 43 Cauchy .................................................................................................................................. 45 Chi......................................................................................................................................... 47 Chi-Square(d) ........................................................................................................................ 49 Cumulative Ascending ........................................................................................................... 51 Cumulative Descending ......................................................................................................... 53 Dagum .................................................................................................................................. 54 Delaporte .............................................................................................................................. 56 Discrete ................................................................................................................................. 58 Discrete Uniform ................................................................................................................... 60 Error Function ....................................................................................................................... 62 m-Erlang................................................................................................................................ 63 Generalized Error .................................................................................................................. 65 Exponential ........................................................................................................................... 68 Extreme Value Max ............................................................................................................... 70 Extreme Value Min ................................................................................................................ 72 F ............................................................................................................................................ 74 Fatigue Life ............................................................................................................................ 76 Gamma ................................................................................................................................. 78 Geometric ............................................................................................................................. 80 Generalized Extreme Value ................................................................................................... 82 Generalized Logistic .............................................................................................................. 84 Histogram ............................................................................................................................. 86 Hyperbolic Secant ................................................................................................................. 88 HypergeoD ............................................................................................................................ 90 HypergeoM ........................................................................................................................... 92 Hypergeometric .................................................................................................................... 94 Inverse Gaussian ................................................................................................................... 96 Inverse Hypergeometric ........................................................................................................ 98 Johnson Bounded ...............................................................................................................
Recommended publications
  • Arxiv:2004.02679V2 [Math.OA] 17 Jul 2020
    THE FREE TANGENT LAW WIKTOR EJSMONT AND FRANZ LEHNER Abstract. Nevanlinna-Herglotz functions play a fundamental role for the study of infinitely divisible distributions in free probability [11]. In the present paper we study the role of the tangent function, which is a fundamental Herglotz-Nevanlinna function [28, 23, 54], and related functions in free probability. To be specific, we show that the function tan z 1 ´ x tan z of Carlitz and Scoville [17, (1.6)] describes the limit distribution of sums of free commutators and anticommutators and thus the free cumulants are given by the Euler zigzag numbers. 1. Introduction Nevanlinna or Herglotz functions are functions analytic in the upper half plane having non- negative imaginary part. This class has been thoroughly studied during the last century and has proven very useful in many applications. One of the fundamental examples of Nevanlinna functions is the tangent function, see [6, 28, 23, 54]. On the other hand it was shown by by Bercovici and Voiculescu [11] that Nevanlinna functions characterize freely infinitely divisible distributions. Such distributions naturally appear in free limit theorems and in the present paper we show that the tangent function appears in a limit theorem for weighted sums of free commutators and anticommutators. More precisely, the family of functions tan z 1 x tan z ´ arises, which was studied by Carlitz and Scoville [17, (1.6)] in connection with the combinorics of tangent numbers; in particular we recover the tangent function for x 0. In recent years a number of papers have investigated limit theorems for“ the free convolution of probability measures defined by Voiculescu [58, 59, 56].
    [Show full text]
  • 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
    [Show full text]
  • The Poisson Burr X Inverse Rayleigh Distribution and Its Applications
    Journal of Data Science,18(1). P. 56 – 77,2020 DOI:10.6339/JDS.202001_18(1).0003 The Poisson Burr X Inverse Rayleigh Distribution And Its Applications Rania H. M. Abdelkhalek* Department of Statistics, Mathematics and Insurance, Benha University, Egypt. ABSTRACT A new flexible extension of the inverse Rayleigh model is proposed and studied. Some of its fundamental statistical properties are derived. We assessed the performance of the maximum likelihood method via a simulation study. The importance of the new model is shown via three applications to real data sets. The new model is much better than other important competitive models. Keywords: Inverse Rayleigh; Poisson; Simulation; Modeling. * [email protected] Rania H. M. Abdelkhalek 57 1. Introduction and physical motivation The well-known inverse Rayleigh (IR) model is considered as a distribution for a life time random variable (r.v.). The IR distribution has many applications in the area of reliability studies. Voda (1972) proved that the distribution of lifetimes of several types of experimental (Exp) units can be approximated by the IR distribution and studied some properties of the maximum likelihood estimation (MLE) of the its parameter. Mukerjee and Saran (1984) studied the failure rate of an IR distribution. Aslam and Jun (2009) introduced a group acceptance sampling plans for truncated life tests based on the IR. Soliman et al. (2010) studied the Bayesian and non- Bayesian estimation of the parameter of the IR model Dey (2012) mentioned that the IR model has also been used as a failure time distribution although the variance and higher order moments does not exist for it.
    [Show full text]
  • New Proposed Length-Biased Weighted Exponential and Rayleigh Distribution with Application
    CORE Metadata, citation and similar papers at core.ac.uk Provided by International Institute for Science, Technology and Education (IISTE): E-Journals Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.7, 2014 New Proposed Length-Biased Weighted Exponential and Rayleigh Distribution with Application Kareema Abed Al-kadim1 Nabeel Ali Hussein 2 1,2.College of Education of Pure Sciences, University of Babylon/Hilla 1.E-mail of the Corresponding author: [email protected] 2.E-mail: [email protected] Abstract The concept of length-biased distribution can be employed in development of proper models for lifetime data. Length-biased distribution is a special case of the more general form known as weighted distribution. In this paper we introduce a new class of length-biased of weighted exponential and Rayleigh distributions(LBW1E1D), (LBWRD).This paper surveys some of the possible uses of Length - biased distribution We study the some of its statistical properties with application of these new distribution. Keywords: length- biased weighted Rayleigh distribution, length- biased weighted exponential distribution, maximum likelihood estimation. 1. Introduction Fisher (1934) [1] introduced the concept of weighted distributions Rao (1965) [2]developed this concept in general terms by in connection with modeling statistical data where the usual practi ce of using standard distributions were not found to be appropriate, in which various situations that can be modeled by weighted distributions, where non-observe ability of some events or damage caused to the original observation resulting in a reduced value, or in a sampling procedure which gives unequal chances to the units in the original That means the recorded observations cannot be considered as a original random sample.
    [Show full text]
  • A Matrix-Valued Bernoulli Distribution Gianfranco Lovison1 Dipartimento Di Scienze Statistiche E Matematiche “S
    Journal of Multivariate Analysis 97 (2006) 1573–1585 www.elsevier.com/locate/jmva A matrix-valued Bernoulli distribution Gianfranco Lovison1 Dipartimento di Scienze Statistiche e Matematiche “S. Vianelli”, Università di Palermo Viale delle Scienze 90128 Palermo, Italy Received 17 September 2004 Available online 10 August 2005 Abstract Matrix-valued distributions are used in continuous multivariate analysis to model sample data matrices of continuous measurements; their use seems to be neglected for binary, or more generally categorical, data. In this paper we propose a matrix-valued Bernoulli distribution, based on the log- linear representation introduced by Cox [The analysis of multivariate binary data, Appl. Statist. 21 (1972) 113–120] for the Multivariate Bernoulli distribution with correlated components. © 2005 Elsevier Inc. All rights reserved. AMS 1991 subject classification: 62E05; 62Hxx Keywords: Correlated multivariate binary responses; Multivariate Bernoulli distribution; Matrix-valued distributions 1. Introduction Matrix-valued distributions are used in continuous multivariate analysis (see, for exam- ple, [10]) to model sample data matrices of continuous measurements, allowing for both variable-dependence and unit-dependence. Their potentials seem to have been neglected for binary, and more generally categorical, data. This is somewhat surprising, since the natural, elementary representation of datasets with categorical variables is precisely in the form of sample binary data matrices, through the 0-1 coding of categories. E-mail address: [email protected] URL: http://dssm.unipa.it/lovison. 1 Work supported by MIUR 60% 2000, 2001 and MIUR P.R.I.N. 2002 grants. 0047-259X/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmva.2005.06.008 1574 G.
    [Show full text]
  • 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.
    [Show full text]
  • On the Computation of Multivariate Scenario Sets for the Skew-T and Generalized Hyperbolic Families Arxiv:1402.0686V1 [Math.ST]
    On the Computation of Multivariate Scenario Sets for the Skew-t and Generalized Hyperbolic Families Emanuele Giorgi1;2, Alexander J. McNeil3;4 February 5, 2014 Abstract We examine the problem of computing multivariate scenarios sets for skewed distributions. Our interest is motivated by the potential use of such sets in the stress testing of insurance companies and banks whose solvency is dependent on changes in a set of financial risk factors. We define multivariate scenario sets based on the notion of half-space depth (HD) and also introduce the notion of expectile depth (ED) where half-spaces are defined by expectiles rather than quantiles. We then use the HD and ED functions to define convex scenario sets that generalize the concepts of quantile and expectile to higher dimensions. In the case of elliptical distributions these sets coincide with the regions encompassed by the contours of the density function. In the context of multivariate skewed distributions, the equivalence of depth contours and density contours does not hold in general. We consider two parametric families that account for skewness and heavy tails: the generalized hyperbolic and the skew- t distributions. By making use of a canonical form representation, where skewness is completely absorbed by one component, we show that the HD contours of these distributions are near-elliptical and, in the case of the skew-Cauchy distribution, we prove that the HD contours are exactly elliptical. We propose a measure of multivariate skewness as a deviation from angular symmetry and show that it can explain the quality of the elliptical approximation for the HD contours.
    [Show full text]
  • Discrete Probability Distributions Uniform Distribution Bernoulli
    Discrete Probability Distributions Uniform Distribution Experiment obeys: all outcomes equally probable Random variable: outcome Probability distribution: if k is the number of possible outcomes, 1 if x is a possible outcome p(x)= k ( 0 otherwise Example: tossing a fair die (k = 6) Bernoulli Distribution Experiment obeys: (1) a single trial with two possible outcomes (success and failure) (2) P trial is successful = p Random variable: number of successful trials (zero or one) Probability distribution: p(x)= px(1 − p)n−x Mean and variance: µ = p, σ2 = p(1 − p) Example: tossing a fair coin once Binomial Distribution Experiment obeys: (1) n repeated trials (2) each trial has two possible outcomes (success and failure) (3) P ith trial is successful = p for all i (4) the trials are independent Random variable: number of successful trials n x n−x Probability distribution: b(x; n,p)= x p (1 − p) Mean and variance: µ = np, σ2 = np(1 − p) Example: tossing a fair coin n times Approximations: (1) b(x; n,p) ≈ p(x; λ = pn) if p ≪ 1, x ≪ n (Poisson approximation) (2) b(x; n,p) ≈ n(x; µ = pn,σ = np(1 − p) ) if np ≫ 1, n(1 − p) ≫ 1 (Normal approximation) p Geometric Distribution Experiment obeys: (1) indeterminate number of repeated trials (2) each trial has two possible outcomes (success and failure) (3) P ith trial is successful = p for all i (4) the trials are independent Random variable: trial number of first successful trial Probability distribution: p(x)= p(1 − p)x−1 1 2 1−p Mean and variance: µ = p , σ = p2 Example: repeated attempts to start
    [Show full text]
  • STATS8: Introduction to Biostatistics 24Pt Random Variables And
    STATS8: Introduction to Biostatistics Random Variables and Probability Distributions Babak Shahbaba Department of Statistics, UCI Random variables • In this lecture, we will discuss random variables and their probability distributions. • Formally, a random variable X assigns a numerical value to each possible outcome (and event) of a random phenomenon. • For instance, we can define X based on possible genotypes of a bi-allelic gene A as follows: 8 < 0 for genotype AA; X = 1 for genotype Aa; : 2 for genotype aa: • Alternatively, we can define a random, Y , variable this way: 0 for genotypes AA and aa; Y = 1 for genotype Aa: Random variables • After we define a random variable, we can find the probabilities for its possible values based on the probabilities for its underlying random phenomenon. • This way, instead of talking about the probabilities for different outcomes and events, we can talk about the probability of different values for a random variable. • For example, suppose P(AA) = 0:49, P(Aa) = 0:42, and P(aa) = 0:09. • Then, we can say that P(X = 0) = 0:49, i.e., X is equal to 0 with probability of 0.49. • Note that the total probability for the random variable is still 1. Random variables • The probability distribution of a random variable specifies its possible values (i.e., its range) and their corresponding probabilities. • For the random variable X defined based on genotypes, the probability distribution can be simply specified as follows: 8 < 0:49 for x = 0; P(X = x) = 0:42 for x = 1; : 0:09 for x = 2: Here, x denotes a specific value (i.e., 0, 1, or 2) of the random variable.
    [Show full text]
  • Benford's Law As a Logarithmic Transformation
    Benford’s Law as a Logarithmic Transformation J. M. Pimbley Maxwell Consulting, LLC Abstract We review the history and description of Benford’s Law - the observation that many naturally occurring numerical data collections exhibit a logarithmic distribution of digits. By creating numerous hypothetical examples, we identify several cases that satisfy Benford’s Law and several that do not. Building on prior work, we create and demonstrate a “Benford Test” to determine from the functional form of a probability density function whether the resulting distribution of digits will find close agreement with Benford’s Law. We also discover that one may generalize the first-digit distribution algorithm to include non-logarithmic transformations. This generalization shows that Benford’s Law rests on the tendency of the transformed and shifted data to exhibit a uniform distribution. Introduction What the world calls “Benford’s Law” is a marvelous mélange of mathematics, data, philosophy, empiricism, theory, mysticism, and fraud. Even with all these qualifiers, one easily describes this law and then asks the simple questions that have challenged investigators for more than a century. Take a large collection of positive numbers which may be integers or real numbers or both. Focus only on the first (non-zero) digit of each number. Count how many numbers of the large collection have each of the nine possibilities (1-9) as the first digit. For typical J. M. Pimbley, “Benford’s Law as a Logarithmic Transformation,” Maxwell Consulting Archives, 2014. number collections – which we’ll generally call “datasets” – the first digit is not equally distributed among the values 1-9.
    [Show full text]
  • A Formulation for Continuous Mixtures of Multivariate Normal Distributions
    A formulation for continuous mixtures of multivariate normal distributions Reinaldo B. Arellano-Valle Adelchi Azzalini Departamento de Estadística Dipartimento di Scienze Statistiche Pontificia Universidad Católica de Chile Università di Padova Chile Italia 31st March 2020 Abstract Several formulations have long existed in the literature in the form of continuous mixtures of normal variables where a mixing variable operates on the mean or on the variance or on both the mean and the variance of a multivariate normal variable, by changing the nature of these basic constituents from constants to random quantities. More recently, other mixture-type constructions have been introduced, where the core random component, on which the mixing operation oper- ates, is not necessarily normal. The main aim of the present work is to show that many existing constructions can be encompassed by a formulation where normal variables are mixed using two univariate random variables. For this formulation, we derive various general properties. Within the proposed framework, it is also simpler to formulate new proposals of parametric families and we provide a few such instances. At the same time, the exposition provides a review of the theme of normal mixtures. Key-words: location-scale mixtures, mixtures of normal distribution. arXiv:2003.13076v1 [math.PR] 29 Mar 2020 1 1 Continuous mixtures of normal distributions In the last few decades, a number of formulations have been put forward, in the context of distribution theory, where a multivariate normal variable represents the basic constituent but with the superpos- ition of another random component, either in the sense that the normal mean value or the variance matrix or both these components are subject to the effect of another random variable of continuous type.
    [Show full text]
  • 1988: Logarithmic Series Distribution and Its Use In
    LOGARITHMIC SERIES DISTRIBUTION AND ITS USE IN ANALYZING DISCRETE DATA Jeffrey R, Wilson, Arizona State University Tempe, Arizona 85287 1. Introduction is assumed to be 7. for any unit and any trial. In a previous paper, Wilson and Koehler Furthermore, for ~ach unit the m trials are (1988) used the generalized-Dirichlet identical and independent so upon summing across multinomial model to account for extra trials X. = (XI~. X 2 ...... X~.)', the vector of variation. The model allows for a second order counts ~r the j th ~nit has ~3multinomial of pairwise correlation among units, a type of distribution with probability vector ~ = (~1' assumption found reasonable in some biological 72 ..... 71) ' and sample size m. Howe~er, data. In that paper, the two-way crossed responses given by the J units at a particular generalized Dirichlet Multinomial model was used trial may be correlated, producing a set of J to analyze repeated measure on the categorical correlated multinomial random vectors, X I, X 2, preferences of insurance customers. The number • o.~ X • of respondents was assumed to be fixed and Ta~is (1962) developed a model for this known. situation, which he called the generalized- In this paper a generalization of the model multinomial distribution in which a single is made allowing the number of respondents m, to parameter P, is used to reflect the common be random. Thus both the number of units m, and dependency between any two of the dependent the underlying probability vector are allowed to multinomial random vectors. The distribution of vary. The model presented here uses the m logarithmic series distribution to account for the category total Xi3.
    [Show full text]