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 ...............................................................................................................