ARNOLD AND STRAUSS S BIVARIATE EXPONENTIAL DISTRIBUTION PRODUCTS AND RATIOS. Saralees Nadarajah and Dongseok Choi (Received February 2005)
|
|
- Caitlin Garrison
- 5 years ago
- Views:
Transcription
1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 35 6), ARNOLD AND STRAUSS S BIVARIATE EXPONENTIAL DISTRIBUTION PRODUCTS AND RATIOS Saralees Nadarajah and Dongseok Choi Received February 5) Abstract. We derive the distributions of W = X/X + Y ) equivalently, X/Y ) and P = XY and the the corresponding moment properties when X and Y follow Arnold and Strauss s bivariate exponential distribution. The expressions turn out to involve several special functions. We also provide extensive tabulations of the percentage points associated with the two distributions. These tables obtained using intensive computing power will be of use to practitioners of the bivariate exponential distribution. 1. Introduction For a bivariate random vector X, Y ), the distributions of the ratios X/X + Y ) and X/Y and the product XY are of interest in problems in biological and physical sciences, econometrics, classification, and ranking and selection. 1) Examples of the use of the ratio of random variables include Mendelian inheritance ratios in genetics, mass to energy ratios in nuclear physics, target to control precipitation in meteorology, and inventory ratios in economics. The distributions of X/X + Y ) and X/Y have been studied by several authors especially when X and Y are independent random variables and come from the same family. For instance, see Marsaglia 1965) and Korhonen and Narula 1989) for normal family, Press 1969) for Student s t family, Basu and Lochner 1971) for Weibull family, Shcolnick 1985) for stable family, Hawkins and Han 1986) for non central chi squared family, Provost 1989b) for gamma family, and Pham Gia ) for beta family. ) As an example of the use of the product of random variables in Physics, Sornette 1998) mentions:... To mimic system size limitation, Takayasu, Sato, and Takayasu introduced a threshold x c... and found a stretched exponential truncating the power law pdf beyond x c. Frisch and Sornette recently developed a theory of extreme deviations generalizing the central limit theorem which, when applied to multiplication of random variables, predicts the generic presence of stretched exponential pdfs. The problem thus boils down to determining the tail of the pdf for a product of random variables... The distribution of XY has been studied by several authors especially when X and Y are independent random variables and come from the same family. For instance, see Sakamoto 1943) for uniform family, Harter 1951) and 1991 Mathematics Subject Classification 33C9, 6E99.
2 19 SARALEES NADARAJAH AND DONGSEOK CHOI Wallgren 198) for Student s t family, Springer and Thompson 197) for normal family, Stuart 196) and Podolski 197) for gamma family, Steece 1976), Bhargava and Khatri 1981) and Tang and Gupta 1984) for beta family, AbuSalih 1983) for power function family, and Malik and Trudel 1986) for exponential family see also Rathie and Rohrer 1987) for a comprehensive review of known results ). There is relatively little work of the above kind when X and Y are correlated random variables. Some of the known work for ratios include Hinkley 1969) for bivariate normal family, Kappenman 1971) for bivariate t family, and Lee et al 1979) for bivariate gamma family. The only work known to the author for products is that by Garg et al ) for Dirichlet family.in this paper, we consider the distributions of W = X/X + Y ) equivalently, X/Y ) and P = XY when X and Y are correlated exponential random variables with the joint pdf f x, y) = K exp { ax + by + cxy)} 1) for x >, y >, a >, b > and c >, where K = Ka, b, c) is the normalizing constant given by ) 1 ab = c exp Ei ab ) K c c and Ei ) is denotes the exponential integral function defined by Ei x) = x expt) dt. t This distribution is due to Arnold and Strauss 1988) and is known as the conditionally specified bivariate exponential distribution. The marginal pdf of X and the conditional pdf of X given Y = y are and f X x) = K exp ax) b + cx f X Y x y) = a + cy) exp { a + cy)x}, respectively. As often with the exponential distribution, 1) has applications in reliability studies. Inaba and Shirahata 1986) fitted 1) to data on white blood counts and survival times of patients who died of acute myelogenous leukemia Gross and Clark, 1975), comparing it with the bivariate normal distribution. Furthermore, note that 1) belongs to the exponential family. Thus, by Lemma 8 in Lehmann 1997), one can obtain confidence intervals for a, b and c by conditioning on part of the sufficient statistic when sampling from 1). The paper is organized as follows. In Sections and 3, we derive explicit expressions for the pdfs, cdfs and moments of W = X/X + Y ) equivalently, X/Y ) and P = XY. In Section 4, we provide extensive tabulations of the associated percentage points, obtained by means of intensive computing power. These values will be of use to the practitioners of the bivariate gamma distribution.
3 PRODUCTS AND RATIOS 191 The calculations of this paper involve several special functions, including the modified Bessel function of the third kind defined by πx ν K ν x) = ν exp xt) t 1 ) ν 1/ dt, Γ ν + 1/) and, the Kummer function defined by Ψa, b; x) = 1 Γa) We also need the following important lemmas. t a t) b a 1 exp xt)dt. Lemma 1.1 Equation ), Prudnikov et al., 1986, volume 1). For p >, x n exp px qx ) π dx = 1) n n { ) q p q n exp Φ q )}, 4p p where Φ ) denotes the cumulative distribution function of the standard normal distribution. Lemma 1. Equation ), Prudnikov et al., 1986, volume 1). For p > and q >, x α 1 exp px q/x) dx = q/p) α/ K α pq). Lemma 1.3 Equation.3.6.9), Prudnikov et al., 1986, volume 1). For α > and p >, x α 1 exp px) x + z) ρ dx = Γα)z α ρ Ψ α, α + 1 ρ; pz). The properties of the above special functions can be found in Prudnikov et al. 1986) and Gradshteyn and Ryzhik ).. and CDF Theorems.1 to. derive the pdfs of W = X/X + Y ) and P = XY when X and Y are distributed according to 1). The corresponding cdf for R = X/Y is also given in Theorem.1. Theorem.1. If X and Y are jointly distributed according to 1) then the pdf of W = X/X + Y ) is given by { } K ΦC) f W w) = 4c A φc) ca ) for < w < 1, where A = w1 w), B = aw + b1 w), C = B/{ ca} and φ ) denotes the pdf of the standard normal distribution. Equivalently, the pdf of R = X/Y is given by { } K ΦC) f R r) = 4c A 1 + r) φc) ca 3)
4 19 SARALEES NADARAJAH AND DONGSEOK CHOI for < r <, where A = r/{1 + r) }, B = ar + b)/1 + r) and C = B/{ ca}. Furthermore, the cdf of R = X/Y can be expressed in the series form F R r) = 1 K { ar + b) } a cr exp c m 4cr a) m ) m n m ar + b) 1) n n cr) h m n/ n where h n x) is given by m= h n x) = x n= z n exp ) z dz. Proof. From 1), the joint pdf of S, W ) = X + Y, X/S) becomes fs, w) = Ks exp [ s {aw + b1 w)} cs w1 w) ]. Thus, the pdf of W can be written as f W w) = K s exp [ s {aw + b1 w)} cs w1 w) ] ds. ) ar + b, cr The result in ) follows by using Lemma 1.1 to calculate the above integral. The result in 3) follows by noting that if f W ) denotes the pdf of W then that of R is given 1 + r) f W r/1 + r)). The cdf of R can be calculated as follows: P R < r) = 1 P X > ry ) = 1 K ry exp { ax + by + cxy)} dxdy exp { ar + b)y cry } = 1 K dy a + cy { ar + b) } { 1 = 1 K exp 4cr a + cy exp cr y + ar + b ) } dy cr = 1 K { ar + b) } a exp 1) m c ) m { y m exp cr 4cr a m= = 1 K { ar + b) } a cr exp 1) m c ) m 4cr a m= { z ar+b)/ ar + b } m ) exp z dz cr cr cr = 1 K { ar + b) } a cr exp 1) m c ) m n ) m ar + b 4cr a n cr m= n= ar+b)/ z n exp z cr 4) y + ar + b ) } dy cr ) m n 1 cr ) n ) dz, 5) where we have substituted z = cr{y + ar + b)/cr)}. The result in 4) follows immediately from 5).
5 PRODUCTS AND RATIOS 193 Checking convergence of the infinite series in 4) is an open problem. Note also that the terms h n ) have been widely used elsewhere in statistics and recursive formulas are available for their computation. Theorem.. If X and Y are jointly distributed according to 1) then f P p) = K exp cp)k ) abp for < p <. Proof. From 1), the joint pdf of X, P ) = X, XY ) becomes fx, p) = Kx 1 exp cp) exp ax bp ). x Thus, the pdf of P can be written as f P p) = Kp β 1 exp cp) 6) x 1 exp ax bp ) dx. 7) x The result of the theorem follows by using Lemma 1. to calculate the above integral. a) b) w w c) d) w w Figure 1. Plots of the pdf of ) for a): c =.1; b): c = 1; c): c = 3; and, d): c = 5. The four curves in each plot are: the solid curve a = 1, b = 1), the curve
6 194 SARALEES NADARAJAH AND DONGSEOK CHOI of lines a = 1, b = 3), the curve of dots a = 3, b = 1), and the curve of lines and dots a = 3, b = 3). a) b) p p c) d) p p Figure. Plots of the pdf of 6) for a): c =.1; b): c = 1; c): c = 3; and, d): c = 5. The four curves in each plot are: the solid curve a = 1, b = 1), the curve of lines a = 1, b = 3), the curve of dots a = 3, b = 1), and the curve of lines and dots a = 3, b = 3). Figures 1 to illustrate the shape of the pdfs ) and 6) for selected values of a, b and c. Each plot contains four curves corresponding to selected values of a and b. The effect of the parameters is evident. 3. Moments Here, we derive the moments of P = XY when X and Y are distributed according to 1). We need the following lemma. Lemma 3.1. If X and Y are jointly distributed according to 1) then m!n!b m n c m Ψ m + 1, m n + 1; ab ) E X m Y n c ) = Ψ 1, 1; ab ) c for m 1 and n 1.
7 PRODUCTS AND RATIOS 195 Proof. One can write E X m Y n ) = K = K = K = Kn! x m y n fx, y)dydx, x m y n exp { ax + by + cxy)} dydx x m exp ax) = Km!n!b m n c m+1) Ψ y n exp { b + cx)y} dydx x m exp ax)b + cx) n+1) dx m + 1, m n + 1; ab ), c where the last equality follows by application of Lemma 1.3. The normalizing constant K is the reciprocal of same integral above when m = and n =. The result of the lemma is immediate. The moments of P = XY are now an immediate consequence of this lemma. Theorem 3.. If X and Y are jointly distributed according to 1) then n!n!c n Ψ n + 1, 1; ab ) E P n c ) = Ψ 1, 1; ab ) c for n 1. Proof. Follows by writing EP n ) = EX n Y n ) and applying the lemma with m = n. 4. Percentiles In this section, we provide extensive tabulations of the percentiles of the distributions of W and P. These percentiles are computed numerically by solving the equations and wα pα f W w)dw = α f P p)dp = α, where f W w) and f P p) are given by ) and 6), respectively. Evidently, this involves computation of the complementary error and Bessel functions and routines for this are widely available. We used the functions erfc ) and BesselK ) in the algebraic manipulation package, MAPLE. The percentiles are given for α =.9,.95,.975,.99,.995,.999,.9995, a = 1, b = 1 and c =.1,.,..., 5. c Percentage points, w α, for W = X/X + Y ) α
8 196 SARALEES NADARAJAH AND DONGSEOK CHOI
9 PRODUCTS AND RATIOS 197 Percentage points, p α, for P = XY c α
10 198 SARALEES NADARAJAH AND DONGSEOK CHOI Similar tabulations could be easily derived for other values of a, b and c. We hope these numbers will be of use to the practitioners of the bivariate exponential distribution see Section 1). Acknowledgements. The authors would like to thank the referee and the editor for carefully reading the paper and for their great help in improving the paper. References 1. M.S. Abu-Salih, Distributions of the product and the quotient of power-function random variables, Arab Journal of Mathematics, ), B.C. Arnold and D.J. Strauss, Bivariate distributions with exponential conditionals, Journal of the American Statistical Association, ), A.P. Basu and R.H. Lochner, On the distribution of the having generalized life distributions, Technometrics, ), R.P. Bhargava and C.G. Khatri, The distribution of product of independent beta random variables with application to multivariate analysis, Annals of the Institute of Statistical Mathematics, ), M. Garg, V. Katta and M.K. Gupta, The distribution of the products of powers of generalized Dirichlet components, Kyungpook Mathematical Journal, 4 ), I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, sixth edition), San Diego, Academic Press,. 7. A.J. Gross and V.A. Clark, Survival Distributions: Reliability Applications in the Biomedical Sciences, New York, John Wiley and Sons, H.L. Harter, On the distribution of Wald s classification statistic, Annals of Mathematical Statistics, 1951), D.L. Hawkins and C.-P. Han, Bivariate distributions noncentral chi square random variables, Communications in Statistics Theory and Methods, ), D.V. Hinkley, On the ratio of two correlated normal random variables, Biometrika, ), T. Inaba and S. Shirahata, Measures of dependence in normal models and exponential models by information gain, Biometrika, ), R.F. Kappenman, A note on the multivariate t ratio distribution, Annals of Mathematical Statistics, ), P.J. Korhonen and S.C. Narula, The probability distribution of the ratio of the absolute values of two normal variables, Journal of Statistical Computation and Simulation, ), R.Y. Lee, B.S. Holland and J.A. Flueck, Distribution of a ratio of correlated gamma random variables, SIAM Journal on Applied Mathematics, ), E.L. Lehmann, Testing Statistical Hypotheses, New York, Springer Verlag, 1997.
11 PRODUCTS AND RATIOS H.J. Malik and R. Trudel, Probability density function of the product and quotient of two correlated exponential random variables, Canadian Mathematical Bulletin, ), G. Marsaglia, Ratios of normal variables and ratios of sums of uniform variables, Journal of the American Statistical Association, ), T. Pham Gia, Distributions of the ratios of independent beta variables and applications, Communications in Statistics Theory and Methods, 9 ), H. Podolski, The distribution of a product of n independent random variables with generalized gamma distribution, Demonstratio Mathematica, 4 197), S.J. Press, The t ratio distribution, Journal of the American Statistical Association, ), S.B. Provost, On the distribution of the ratio of powers of sums of gamma random variables, Pakistan Journal Statistics, ), A.P. Prudnikov, Y.A. Brychkov and O.I. Marichev, Integrals and Series, Volumes 1, and 3), Amsterdam, Gordon and Breach Science Publishers, P.N. Rathie and H.G. Rohrer, The exact distribution of products of independent random variables, Metron, ), H. Sakamoto, On the distributions of the product and the quotient of the independent and uniformly distributed random variables, Tohoku Mathematical Journal, ), S.M. Shcolnick, On the ratio of independent stable random variables, Stability Problems for Stochastic Models, Uzhgorod, 1984, pp , Lecture Notes in Mathematics, Vol. 1155, Springer, Berlin, D. Sornette, Multiplicative processes and power laws, Physical Review E, ), M.D. Springer and W.E. Thompson, The distribution of products of beta, gamma and Gaussian random variables, SIAM Journal on Applied Mathematics, ), B.M. Steece, On the exact distribution for the product of two independent beta distributed random variables, Metron, ), A. Stuart, Gamma distributed products of independent random variables, Biometrika, ), J. Tang and A.K. Gupta, On the distribution of the product of independent beta random variables, Statistics and Probability Letters, 1984), C.M. Wallgren, The distribution of the product of two correlated t variates, Journal of the American Statistical Association, ), Saralees Nadarajah School of Mathematics University of Manchester Oxford Road Manchester M13 9PL UNITED KINGDOM Saralees.Nadarajah@manchester.ac.uk Dongseok Choi Department of Public Health and Preventive Medicine Oregon Health and Science University Portland Oregon 9739 USA choid@ohsu.edu
On the Ratio of Rice Random Variables
JIRSS 9 Vol. 8, Nos. 1-, pp 61-71 On the Ratio of Rice Random Variables N. B. Khoolenjani 1, K. Khorshidian 1, 1 Departments of Statistics, Shiraz University, Shiraz, Iran. n.b.khoolenjani@gmail.com Departments
More informationThe Distributions of Sums, Products and Ratios of Inverted Bivariate Beta Distribution 1
Applied Mathematical Sciences, Vol. 2, 28, no. 48, 2377-2391 The Distributions of Sums, Products and Ratios of Inverted Bivariate Beta Distribution 1 A. S. Al-Ruzaiza and Awad El-Gohary 2 Department of
More informationProducts and Ratios of Two Gaussian Class Correlated Weibull Random Variables
Products and Ratios of Two Gaussian Class Correlated Weibull Random Variables Petros S. Bithas, Nikos C. Sagias 2, Theodoros A. Tsiftsis 3, and George K. Karagiannidis 3 Electrical and Computer Engineering
More informationOn the Ratio of Inverted Gamma Variates
AUSTRIAN JOURNAL OF STATISTICS Volume 36 2007), Number 2, 153 159 On the Ratio of Inerted Gamma Variates M. Masoom Ali 1, Manisha Pal 2, and Jungsoo Woo 3 Department of Mathematical Sciences, Ball State
More informationEXPLICIT EXPRESSIONS FOR MOMENTS OF χ 2 ORDER STATISTICS
Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 3 (28), No. 3, pp. 433-444 EXPLICIT EXPRESSIONS FOR MOMENTS OF χ 2 ORDER STATISTICS BY SARALEES NADARAJAH Abstract Explicit closed
More informationSolutions to the recurrence relation u n+1 = v n+1 + u n v n in terms of Bell polynomials
Volume 31, N. 2, pp. 245 258, 2012 Copyright 2012 SBMAC ISSN 0101-8205 / ISSN 1807-0302 (Online) www.scielo.br/cam Solutions to the recurrence relation u n+1 = v n+1 + u n v n in terms of Bell polynomials
More informationA Marshall-Olkin Gamma Distribution and Process
CHAPTER 3 A Marshall-Olkin Gamma Distribution and Process 3.1 Introduction Gamma distribution is a widely used distribution in many fields such as lifetime data analysis, reliability, hydrology, medicine,
More informationChapter 4 Multiple Random Variables
Review for the previous lecture Definition: n-dimensional random vector, joint pmf (pdf), marginal pmf (pdf) Theorem: How to calculate marginal pmf (pdf) given joint pmf (pdf) Example: How to calculate
More informationDistribution of the Ratio of Normal and Rice Random Variables
Journal of Modern Applied Statistical Methods Volume 1 Issue Article 7 11-1-013 Distribution of the Ratio of Normal and Rice Random Variables Nayereh B. Khoolenjani Uniersity of Isfahan, Isfahan, Iran,
More informationStatistics of Correlated Generalized Gamma Variates
Statistics of Correlated Generalized Gamma Variates Petros S. Bithas 1 and Theodoros A. Tsiftsis 2 1 Department of Electrical & Computer Engineering, University of Patras, Rion, GR-26500, Patras, Greece,
More informationComputation of Signal to Noise Ratios
MATCH Communications in Mathematical in Computer Chemistry MATCH Commun. Math. Comput. Chem. 57 7) 15-11 ISS 34-653 Computation of Signal to oise Ratios Saralees adarajah 1 & Samuel Kotz Received May,
More informationResearch Article The Laplace Likelihood Ratio Test for Heteroscedasticity
International Mathematics and Mathematical Sciences Volume 2011, Article ID 249564, 7 pages doi:10.1155/2011/249564 Research Article The Laplace Likelihood Ratio Test for Heteroscedasticity J. Martin van
More informationContinuous Random Variables
1 / 24 Continuous Random Variables Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay February 27, 2013 2 / 24 Continuous Random Variables
More informationOn the product of independent Generalized Gamma random variables
On the product of independent Generalized Gamma random variables Filipe J. Marques Universidade Nova de Lisboa and Centro de Matemática e Aplicações, Portugal Abstract The product of independent Generalized
More informationMultiple Random Variables
Multiple Random Variables This Version: July 30, 2015 Multiple Random Variables 2 Now we consider models with more than one r.v. These are called multivariate models For instance: height and weight An
More informationNotes on a skew-symmetric inverse double Weibull distribution
Journal of the Korean Data & Information Science Society 2009, 20(2), 459 465 한국데이터정보과학회지 Notes on a skew-symmetric inverse double Weibull distribution Jungsoo Woo 1 Department of Statistics, Yeungnam
More information2 Functions of random variables
2 Functions of random variables A basic statistical model for sample data is a collection of random variables X 1,..., X n. The data are summarised in terms of certain sample statistics, calculated as
More informationA GENERALIZED LOGISTIC DISTRIBUTION
A GENERALIZED LOGISTIC DISTRIBUTION SARALEES NADARAJAH AND SAMUEL KOTZ Received 11 October 2004 A generalized logistic distribution is proposed, based on the fact that the difference of two independent
More informationDeccan Education Society s FERGUSSON COLLEGE, PUNE (AUTONOMOUS) SYLLABUS UNDER AUTOMONY. SECOND YEAR B.Sc. SEMESTER - III
Deccan Education Society s FERGUSSON COLLEGE, PUNE (AUTONOMOUS) SYLLABUS UNDER AUTOMONY SECOND YEAR B.Sc. SEMESTER - III SYLLABUS FOR S. Y. B. Sc. STATISTICS Academic Year 07-8 S.Y. B.Sc. (Statistics)
More informationMAS223 Statistical Inference and Modelling Exercises
MAS223 Statistical Inference and Modelling Exercises The exercises are grouped into sections, corresponding to chapters of the lecture notes Within each section exercises are divided into warm-up questions,
More informationA BIVARIATE GENERALISATION OF GAMMA DISTRIBUTION
A BIVARIATE GENERALISATION OF GAMMA DISTRIBUTION J. VAN DEN BERG, J.J.J. ROUX and A. BEKKER Department of Statistics, Faculty of Natural and Agricultural Sciences, University of Pretoria, Pretoria,, South
More informationLecture 3. Probability - Part 2. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza. October 19, 2016
Lecture 3 Probability - Part 2 Luigi Freda ALCOR Lab DIAG University of Rome La Sapienza October 19, 2016 Luigi Freda ( La Sapienza University) Lecture 3 October 19, 2016 1 / 46 Outline 1 Common Continuous
More informationOn the Expected Absolute Value of a Bivariate Normal Distribution
Journal of Statistical Theory and Applications Volume, Number 4, 0, pp. 37-377 ISSN 538-7887 On the Expected Absolute Value of a Bivariate Normal Distribution S. Reza H. Shojaie, Mina Aminghafari and Adel
More informationMath 416 Lecture 2 DEFINITION. Here are the multivariate versions: X, Y, Z iff P(X = x, Y = y, Z =z) = p(x, y, z) of X, Y, Z iff for all sets A, B, C,
Math 416 Lecture 2 DEFINITION. Here are the multivariate versions: PMF case: p(x, y, z) is the joint Probability Mass Function of X, Y, Z iff P(X = x, Y = y, Z =z) = p(x, y, z) PDF case: f(x, y, z) is
More informationOn q-gamma Distributions, Marshall-Olkin q-gamma Distributions and Minification Processes
CHAPTER 4 On q-gamma Distributions, Marshall-Olkin q-gamma Distributions and Minification Processes 4.1 Introduction Several skewed distributions such as logistic, Weibull, gamma and beta distributions
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections Chapter 3 - continued 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions
More informationMultivariate Random Variable
Multivariate Random Variable Author: Author: Andrés Hincapié and Linyi Cao This Version: August 7, 2016 Multivariate Random Variable 3 Now we consider models with more than one r.v. These are called multivariate
More informationIntroduction to Probability and Stocastic Processes - Part I
Introduction to Probability and Stocastic Processes - Part I Lecture 2 Henrik Vie Christensen vie@control.auc.dk Department of Control Engineering Institute of Electronic Systems Aalborg University Denmark
More informationSTAT 515 MIDTERM 2 EXAM November 14, 2018
STAT 55 MIDTERM 2 EXAM November 4, 28 NAME: Section Number: Instructor: In problems that require reasoning, algebraic calculation, or the use of your graphing calculator, it is not sufficient just to write
More informationSlides 8: Statistical Models in Simulation
Slides 8: Statistical Models in Simulation Purpose and Overview The world the model-builder sees is probabilistic rather than deterministic: Some statistical model might well describe the variations. An
More informationMA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems
MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
More informationEE4601 Communication Systems
EE4601 Communication Systems Week 2 Review of Probability, Important Distributions 0 c 2011, Georgia Institute of Technology (lect2 1) Conditional Probability Consider a sample space that consists of two
More informationA New Class of Positively Quadrant Dependent Bivariate Distributions with Pareto
International Mathematical Forum, 2, 27, no. 26, 1259-1273 A New Class of Positively Quadrant Dependent Bivariate Distributions with Pareto A. S. Al-Ruzaiza and Awad El-Gohary 1 Department of Statistics
More informationPARAMETER ESTIMATION FOR THE LOG-LOGISTIC DISTRIBUTION BASED ON ORDER STATISTICS
PARAMETER ESTIMATION FOR THE LOG-LOGISTIC DISTRIBUTION BASED ON ORDER STATISTICS Authors: Mohammad Ahsanullah Department of Management Sciences, Rider University, New Jersey, USA ahsan@rider.edu) Ayman
More information3. Probability and Statistics
FE661 - Statistical Methods for Financial Engineering 3. Probability and Statistics Jitkomut Songsiri definitions, probability measures conditional expectations correlation and covariance some important
More informationMultivariate Distribution Models
Multivariate Distribution Models Model Description While the probability distribution for an individual random variable is called marginal, the probability distribution for multiple random variables is
More informationStochastic Comparisons of Weighted Sums of Arrangement Increasing Random Variables
Portland State University PDXScholar Mathematics and Statistics Faculty Publications and Presentations Fariborz Maseeh Department of Mathematics and Statistics 4-7-2015 Stochastic Comparisons of Weighted
More informationASM Study Manual for Exam P, Second Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA Errata
ASM Study Manual for Exam P, Second Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA (krzysio@krzysio.net) Errata Effective July 5, 3, only the latest edition of this manual will have its errata
More informationChapter 3 sections. SKIP: 3.10 Markov Chains. SKIP: pages Chapter 3 - continued
Chapter 3 sections 3.1 Random Variables and Discrete Distributions 3.2 Continuous Distributions 3.3 The Cumulative Distribution Function 3.4 Bivariate Distributions 3.5 Marginal Distributions 3.6 Conditional
More informationA Skewed Look at Bivariate and Multivariate Order Statistics
A Skewed Look at Bivariate and Multivariate Order Statistics Prof. N. Balakrishnan Dept. of Mathematics & Statistics McMaster University, Canada bala@mcmaster.ca p. 1/4 Presented with great pleasure as
More informationChapter 4 Multiple Random Variables
Review for the previous lecture Theorems and Examples: How to obtain the pmf (pdf) of U = g ( X Y 1 ) and V = g ( X Y) Chapter 4 Multiple Random Variables Chapter 43 Bivariate Transformations Continuous
More informationn! (k 1)!(n k)! = F (X) U(0, 1). (x, y) = n(n 1) ( F (y) F (x) ) n 2
Order statistics Ex. 4. (*. Let independent variables X,..., X n have U(0, distribution. Show that for every x (0,, we have P ( X ( < x and P ( X (n > x as n. Ex. 4.2 (**. By using induction or otherwise,
More informationChapter 5 continued. Chapter 5 sections
Chapter 5 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationBurr Type X Distribution: Revisited
Burr Type X Distribution: Revisited Mohammad Z. Raqab 1 Debasis Kundu Abstract In this paper, we consider the two-parameter Burr-Type X distribution. We observe several interesting properties of this distribution.
More informationSUMS, PRODUCTS, AND RATIOS FOR THE GENERALIZED BIVARIATE PARETO DISTRIBUTION. Saralees Nadarajah and Mariano Ruiz Espejo. Abstract
S. NADARAJAH AND M. R. ESPEJO KODAI MATH. J. 29 (26), 72 83 SUMS, PRODUCTS, AND RATIOS FOR THE GENERALIZED BIVARIATE PARETO DISTRIBUTION Saralees Nadarajah and Mariano Ruiz Espejo Abstract We derive the
More informationBivariate Weibull-power series class of distributions
Bivariate Weibull-power series class of distributions Saralees Nadarajah and Rasool Roozegar EM algorithm, Maximum likelihood estimation, Power series distri- Keywords: bution. Abstract We point out that
More informationSTA2603/205/1/2014 /2014. ry II. Tutorial letter 205/1/
STA263/25//24 Tutorial letter 25// /24 Distribution Theor ry II STA263 Semester Department of Statistics CONTENTS: Examination preparation tutorial letterr Solutions to Assignment 6 2 Dear Student, This
More informationSOLUTIONS TO MATH68181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH68181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question A1 a) The marginal cdfs of F X,Y (x, y) = [1 + exp( x) + exp( y) + (1 α) exp( x y)] 1 are F X (x) = F X,Y (x, ) = [1
More informationarxiv: v1 [math.ca] 3 Aug 2008
A generalization of the Widder potential transform and applications arxiv:88.317v1 [math.ca] 3 Aug 8 Neşe Dernek a, Veli Kurt b, Yılmaz Şimşek b, Osman Yürekli c, a Department of Mathematics, University
More informationMultiple Random Variables
Multiple Random Variables Joint Probability Density Let X and Y be two random variables. Their joint distribution function is F ( XY x, y) P X x Y y. F XY ( ) 1, < x
More informationChapter 5. Statistical Models in Simulations 5.1. Prof. Dr. Mesut Güneş Ch. 5 Statistical Models in Simulations
Chapter 5 Statistical Models in Simulations 5.1 Contents Basic Probability Theory Concepts Discrete Distributions Continuous Distributions Poisson Process Empirical Distributions Useful Statistical Models
More informationTheoretical Properties of Weighted Generalized. Rayleigh and Related Distributions
Theoretical Mathematics & Applications, vol.2, no.2, 22, 45-62 ISSN: 792-9687 (print, 792-979 (online International Scientific Press, 22 Theoretical Properties of Weighted Generalized Rayleigh and Related
More informationRELIABILITY FOR SOME BIVARIATE EXPONENTIAL DISTRIBUTIONS
RELIABILITY FOR SOME BIVARIATE EXPONENTIAL DISTRIBUTIONS SARALEES NADARAJAH AND SAMUEL KOTZ Received 8 January 5; Revised 7 March 5; Accepted June 5 In the area of stress-strength models, there has been
More informationEEL 5544 Noise in Linear Systems Lecture 30. X (s) = E [ e sx] f X (x)e sx dx. Moments can be found from the Laplace transform as
L30-1 EEL 5544 Noise in Linear Systems Lecture 30 OTHER TRANSFORMS For a continuous, nonnegative RV X, the Laplace transform of X is X (s) = E [ e sx] = 0 f X (x)e sx dx. For a nonnegative RV, the Laplace
More informationRandom Variables and Their Distributions
Chapter 3 Random Variables and Their Distributions A random variable (r.v.) is a function that assigns one and only one numerical value to each simple event in an experiment. We will denote r.vs by capital
More informationBayesian inference for the location parameter of a Student-t density
Bayesian inference for the location parameter of a Student-t density Jean-François Angers CRM-2642 February 2000 Dép. de mathématiques et de statistique; Université de Montréal; C.P. 628, Succ. Centre-ville
More informationFormulas for probability theory and linear models SF2941
Formulas for probability theory and linear models SF2941 These pages + Appendix 2 of Gut) are permitted as assistance at the exam. 11 maj 2008 Selected formulae of probability Bivariate probability Transforms
More informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 05 SUBJECT NAME : Probability & Random Process SUBJECT CODE : MA6 MATERIAL NAME : University Questions MATERIAL CODE : JM08AM004 REGULATION : R008 UPDATED ON : Nov-Dec 04 (Scan
More informationWeibull-Gamma composite distribution: An alternative multipath/shadowing fading model
Weibull-Gamma composite distribution: An alternative multipath/shadowing fading model Petros S. Bithas Institute for Space Applications and Remote Sensing, National Observatory of Athens, Metaxa & Vas.
More informationComputer Science, Informatik 4 Communication and Distributed Systems. Simulation. Discrete-Event System Simulation. Dr.
Simulation Discrete-Event System Simulation Chapter 4 Statistical Models in Simulation Purpose & Overview The world the model-builder sees is probabilistic rather than deterministic. Some statistical model
More informationRELIABILITY FOR SOME BIVARIATE BETA DISTRIBUTIONS
RELIABILITY FOR SOME BIVARIATE BETA DISTRIBUTIONS SARALEES NADARAJAH Received 24 June 24 In the area of stress-strength models there has been a large amount of work as regards estimation of the reliability
More informationDistributions of Functions of Random Variables. 5.1 Functions of One Random Variable
Distributions of Functions of Random Variables 5.1 Functions of One Random Variable 5.2 Transformations of Two Random Variables 5.3 Several Random Variables 5.4 The Moment-Generating Function Technique
More informationPasquale Erto 1 University of Naples Federico II P.le V. Tecchio 80, Naples, Italy
A NOTE ON MONITORING RATIOS OF WEIBULL PERCENTILES Pasquale Erto 1 University of Naples Federico II P.le V. Tecchio 80, 80125 Naples, Italy This note introduces a new Bayesian control chart to compare
More informationProbability and Stochastic Processes
Probability and Stochastic Processes A Friendly Introduction Electrical and Computer Engineers Third Edition Roy D. Yates Rutgers, The State University of New Jersey David J. Goodman New York University
More informationRELATIONS FOR MOMENTS OF PROGRESSIVELY TYPE-II RIGHT CENSORED ORDER STATISTICS FROM ERLANG-TRUNCATED EXPONENTIAL DISTRIBUTION
STATISTICS IN TRANSITION new series, December 2017 651 STATISTICS IN TRANSITION new series, December 2017 Vol. 18, No. 4, pp. 651 668, DOI 10.21307/stattrans-2017-005 RELATIONS FOR MOMENTS OF PROGRESSIVELY
More informationTHE BIVARIATE F 3 -BETA DISTRIBUTION. Saralees Nadarajah. 1. Introduction
Commun. Korean Math. Soc. 21 2006, No. 2, pp. 363 374 THE IVARIATE F 3 -ETA DISTRIUTION Saralees Nadarajah Abstract. A new bivariate beta distribution based on the Appell function of the third kind is
More informationMultivariate random variables
DS-GA 002 Lecture notes 3 Fall 206 Introduction Multivariate random variables Probabilistic models usually include multiple uncertain numerical quantities. In this section we develop tools to characterize
More information9.07 Introduction to Probability and Statistics for Brain and Cognitive Sciences Emery N. Brown
9.07 Introduction to Probability and Statistics for Brain and Cognitive Sciences Emery N. Brown I. Objectives Lecture 5: Conditional Distributions and Functions of Jointly Distributed Random Variables
More informationOn the entropy flows to disorder
CHAOS 2009 Charnia, Crete 1-5 June 2009 On the entropy flows to disorder C.T.J. Dodson School of Mathematics University of Manchester, UK Abstract Gamma distributions, which contain the exponential as
More informationSTAT Chapter 5 Continuous Distributions
STAT 270 - Chapter 5 Continuous Distributions June 27, 2012 Shirin Golchi () STAT270 June 27, 2012 1 / 59 Continuous rv s Definition: X is a continuous rv if it takes values in an interval, i.e., range
More informationChapter 5. Chapter 5 sections
1 / 43 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 08 SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : University Questions REGULATION : R03 UPDATED ON : November 07 (Upto N/D 07 Q.P) (Scan the
More informationLikelihood Ratio Criterion for Testing Sphericity from a Multivariate Normal Sample with 2-step Monotone Missing Data Pattern
The Korean Communications in Statistics Vol. 12 No. 2, 2005 pp. 473-481 Likelihood Ratio Criterion for Testing Sphericity from a Multivariate Normal Sample with 2-step Monotone Missing Data Pattern Byungjin
More informationPCMI Introduction to Random Matrix Theory Handout # REVIEW OF PROBABILITY THEORY. Chapter 1 - Events and Their Probabilities
PCMI 207 - Introduction to Random Matrix Theory Handout #2 06.27.207 REVIEW OF PROBABILITY THEORY Chapter - Events and Their Probabilities.. Events as Sets Definition (σ-field). A collection F of subsets
More informationContinuous random variables
Continuous random variables Can take on an uncountably infinite number of values Any value within an interval over which the variable is definied has some probability of occuring This is different from
More informationTHE QUEEN S UNIVERSITY OF BELFAST
THE QUEEN S UNIVERSITY OF BELFAST 0SOR20 Level 2 Examination Statistics and Operational Research 20 Probability and Distribution Theory Wednesday 4 August 2002 2.30 pm 5.30 pm Examiners { Professor R M
More informationn! (k 1)!(n k)! = F (X) U(0, 1). (x, y) = n(n 1) ( F (y) F (x) ) n 2
Order statistics Ex. 4.1 (*. Let independent variables X 1,..., X n have U(0, 1 distribution. Show that for every x (0, 1, we have P ( X (1 < x 1 and P ( X (n > x 1 as n. Ex. 4.2 (**. By using induction
More informationE X A M. Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours. Number of pages incl.
E X A M Course code: Course name: Number of pages incl. front page: 6 MA430-G Probability Theory and Stochastic Processes Date: December 13, 2016 Duration: 4 hours Resources allowed: Notes: Pocket calculator,
More informationThe integrals in Gradshteyn and Ryzhik. Part 10: The digamma function
SCIENTIA Series A: Mathematical Sciences, Vol 7 29, 45 66 Universidad Técnica Federico Santa María Valparaíso, Chile ISSN 76-8446 c Universidad Técnica Federico Santa María 29 The integrals in Gradshteyn
More informationECE Lecture #10 Overview
ECE 450 - Lecture #0 Overview Introduction to Random Vectors CDF, PDF Mean Vector, Covariance Matrix Jointly Gaussian RV s: vector form of pdf Introduction to Random (or Stochastic) Processes Definitions
More information1 Probability and Random Variables
1 Probability and Random Variables The models that you have seen thus far are deterministic models. For any time t, there is a unique solution X(t). On the other hand, stochastic models will result in
More informationOn the entropy flows to disorder
On the entropy flows to disorder School of Mathematics University of Manchester Manchester M 9PL, UK (e-mail: ctdodson@manchester.ac.uk) C.T.J. Dodson Abstract. Gamma distributions, which contain the exponential
More informationASM Study Manual for Exam P, First Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA Errata
ASM Study Manual for Exam P, First Edition By Dr. Krzysztof M. Ostaszewski, FSA, CFA, MAAA (krzysio@krzysio.net) Errata Effective July 5, 3, only the latest edition of this manual will have its errata
More informationIEOR 4701: Stochastic Models in Financial Engineering. Summer 2007, Professor Whitt. SOLUTIONS to Homework Assignment 9: Brownian motion
IEOR 471: Stochastic Models in Financial Engineering Summer 27, Professor Whitt SOLUTIONS to Homework Assignment 9: Brownian motion In Ross, read Sections 1.1-1.3 and 1.6. (The total required reading there
More informationEstimation of Stress-Strength Reliability for Kumaraswamy Exponential Distribution Based on Upper Record Values
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 2, 59-71 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.7210 Estimation of Stress-Strength Reliability for
More informationPseudo-Boolean Functions, Lovász Extensions, and Beta Distributions
Pseudo-Boolean Functions, Lovász Extensions, and Beta Distributions Guoli Ding R. F. Lax Department of Mathematics, LSU, Baton Rouge, LA 783 Abstract Let f : {,1} n R be a pseudo-boolean function and let
More informationIntroduction of Shape/Skewness Parameter(s) in a Probability Distribution
Journal of Probability and Statistical Science 7(2), 153-171, Aug. 2009 Introduction of Shape/Skewness Parameter(s) in a Probability Distribution Rameshwar D. Gupta University of New Brunswick Debasis
More informationUNIT Define joint distribution and joint probability density function for the two random variables X and Y.
UNIT 4 1. Define joint distribution and joint probability density function for the two random variables X and Y. Let and represent the probability distribution functions of two random variables X and Y
More informationSELECTING THE NORMAL POPULATION WITH THE SMALLEST COEFFICIENT OF VARIATION
SELECTING THE NORMAL POPULATION WITH THE SMALLEST COEFFICIENT OF VARIATION Ajit C. Tamhane Department of IE/MS and Department of Statistics Northwestern University, Evanston, IL 60208 Anthony J. Hayter
More informationInformation geometry for bivariate distribution control
Information geometry for bivariate distribution control C.T.J.Dodson + Hong Wang Mathematics + Control Systems Centre, University of Manchester Institute of Science and Technology Optimal control of stochastic
More informationHANDBOOK OF APPLICABLE MATHEMATICS
HANDBOOK OF APPLICABLE MATHEMATICS Chief Editor: Walter Ledermann Volume II: Probability Emlyn Lloyd University oflancaster A Wiley-Interscience Publication JOHN WILEY & SONS Chichester - New York - Brisbane
More informationProbability and Distributions
Probability and Distributions What is a statistical model? A statistical model is a set of assumptions by which the hypothetical population distribution of data is inferred. It is typically postulated
More informationStatistical Methods in Particle Physics
Statistical Methods in Particle Physics Lecture 3 October 29, 2012 Silvia Masciocchi, GSI Darmstadt s.masciocchi@gsi.de Winter Semester 2012 / 13 Outline Reminder: Probability density function Cumulative
More information3 Modeling Process Quality
3 Modeling Process Quality 3.1 Introduction Section 3.1 contains basic numerical and graphical methods. familiar with these methods. It is assumed the student is Goal: Review several discrete and continuous
More informationCopula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011
Copula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011 Outline Ordinary Least Squares (OLS) Regression Generalized Linear Models
More information2 Random Variable Generation
2 Random Variable Generation Most Monte Carlo computations require, as a starting point, a sequence of i.i.d. random variables with given marginal distribution. We describe here some of the basic methods
More informationp(z)
Chapter Statistics. Introduction This lecture is a quick review of basic statistical concepts; probabilities, mean, variance, covariance, correlation, linear regression, probability density functions and
More informationLecture 2: Repetition of probability theory and statistics
Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:
More informationNotes for Math 324, Part 19
48 Notes for Math 324, Part 9 Chapter 9 Multivariate distributions, covariance Often, we need to consider several random variables at the same time. We have a sample space S and r.v. s X, Y,..., which
More informationModified Kolmogorov-Smirnov Test of Goodness of Fit. Catalonia-BarcelonaTECH, Spain
152/304 CoDaWork 2017 Abbadia San Salvatore (IT) Modified Kolmogorov-Smirnov Test of Goodness of Fit G.S. Monti 1, G. Mateu-Figueras 2, M. I. Ortego 3, V. Pawlowsky-Glahn 2 and J. J. Egozcue 3 1 Department
More information