CRC standard probability and Statistics tables and formulae c 2000 by Chapman & Hall/CRC CRC standard probability and Statistics tables and formulae DANIEL ZWILLINGER Rensselaer Polytechnic Institute Troy, New York STEPHEN KOKOSKA Bloomsburg University Bloomsburg, Pennsylvania CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C Library of Congress Cataloging-in-Publication Data Zwillinger, Daniel, 1957CRC standard probability and statistics tables and formulae / Daniel Zwillinger, Stephen Kokoska p cm Includes bibliographical references and index ISBN 1-58488-059-7 (alk paper) Probabilities—Tables Mathematical statistics—Tables I Kokoska, Stephen II Title QA273.3 Z95 1999 519.2′02′1—dc21 99-045786 This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale Specific permission must be obtained in writing from CRC Press LLC for such copying Direct all inquiries to CRC Press LLC, 2000 N.W Corporate Blvd., Boca Raton, Florida 33431 Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe Visit the CRC Press Web site at www.crcpress.com © 2000 by Chapman & Hall/CRC No claim to original U.S Government works International Standard Book Number 1-58488-059-7 Library of Congress Card Number 99-045786 Printed in the United States of America Printed on acid-free paper Preface It has long been the established policy of CRC Press to publish, in handbook form, the most up-to-date, authoritative, logically arranged, and readily usable reference material available This book fills the need in probability and statistics Prior to the preparation of this book the contents of similar books were considered It is easy to fill a statistics reference book with many hundred pages of tables—indeed, some large books contain statistical tables for only a single test The authors of this book focused on the basic principles of statistics We have tried to ensure that each topic had an understandable textual introduction as well as easily understood examples There are more than 80 examples; they usually follow the same format: start with a word problem, interpret the words as a statistical problem, find the solution, interpret the solution in words We have organized this reference in an efficient and useful format We believe both students and researchers will find this reference easy to read and understand Material is presented in a multi-sectional format, with each section containing a valuable collection of fundamental reference material—tabular and expository This Handbook serves as a guide for determining appropriate statistical procedures and interpretation of results We have assembled the most important concepts in probability and statistics, as experienced through our own teaching, research, and work in industry For most topics, concise yet useful tables were created In most cases, the tables were re-generated and verified against existing tables Even very modest statistical software can generate many of the tables in the book—often to more decimal places and for more values of the parameters The values in this book are designed to illustrate the range of possible values and act as a handy reference for the most commonly needed values This book also contains many useful topics from more advanced areas of statistics, but these topics have fewer examples Also included are a large collection of short topics containing many classical results and puzzles Finally, a section on notation used in the book and a comprehensive index are also included c 2000 by Chapman & Hall/CRC In line with the established policy of CRC Press, this Handbook will be kept as current and timely as is possible Revisions and anticipated uses of newer materials and tables will be introduced as the need arises Suggestions for the inclusion of new material in subsequent editions and comments concerning the accuracy of stated information are welcomed If any errata are discovered for this book, they will be posted to http://vesta.bloomu.edu/~skokoska/prast/errata Many people have helped in the preparation of this manuscript The authors are especially grateful to our families who have remained lighthearted and cheerful throughout the process A special thanks to Janet and Kent, and to Joan, Mark, and Jen Daniel Zwillinger zwillinger@alum.mit.edu Stephen Kokoska skokoska@planetx.bloomu.edu ACKNOWLEDGMENTS Plans 6.1–6.6, 6A.1–6A.6, and 13.1–13.5 (appearing on pages 331–337) originally appeared on pages 234–237, 276–279, and 522–523 of W G Cochran and G M Cox, Experimental Designs, Second Edition, John Wiley & Sons, Inc, New York, 1957 Reprinted by permission of John Wiley & Sons, Inc The tables of Bartlett’s critical values (in section 10.6.2) are from D D Dyer and J P Keating, “On the Determination of Critical Values for Bartlett’s Test”, JASA, Volume 75, 1980, pages 313–319 Reprinted with permission from the Journal of American Statistical Association Copyright 1980 by the American Statistical Association All rights reserved The tables of Cochran’s critical values (in section 10.7.1) are from C Eisenhart, M W Hastay, and W A Wallis, Techniques of Statistical Analysis, McGraw-Hill Book Company, 1947, Tables 15.1 and 15.2 (pages 390-391) Reprinted courtesy of The McGraw-Hill Companies The tables of Dunnett’s critical values (in section 12.1.4.5) are from C W Dunnett, “A Multiple Comparison Procedure for Comparing Several Treatments with a Control”, JASA, Volume 50, 1955, pages 1096–1121 Reprinted with permission from the Journal of American Statistical Association Copyright 1980 by the American Statistical Association All rights reserved The tables of Duncan’s critical values (in section 12.1.4.3) are from L Hunter, “Critical Values for Duncan’s New Multiple Range Test”, Biometrics, 1960, Volume 16, pages 671– 685 Reprinted with permission from the Journal of American Statistical Association Copyright 1960 by the American Statistical Association All rights reserved Table 15.1 is reproduced, by permission, from ASTM Manual on Quality Control of Materials, American Society for Testing and Materials, Philadelphia, PA, 1951 The table in section 15.1.2 and much of Chapter 18 originally appeared in D Zwillinger, Standard Mathematical Tables and Formulae, 30th edition, CRC Press, Boca Raton, FL, 1995 Reprinted courtesy of CRC Press, LLC Much of section 17.17 is taken from the URL http://members.aol.com/johnp71/javastat.html Permission courtesy of John C Pezzullo c 2000 by Chapman & Hall/CRC Contents Introduction 1.1 Background 1.2 Data sets 1.3 Summarizing Data 2.1 Tabular and graphical procedures 2.2 References Numerical summary measures Probability 3.1 Algebra of sets 3.2 Combinatorial methods 3.3 3.4 Probability Random variables 3.5 3.6 3.7 Mathematical expectation Multivariate distributions Inequalities 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Functions of Random Variables Finding the probability distribution Sums of random variables Sampling distributions Finite population Theorems Order statistics Range and studentized range © 2000 by Chapman & Hall/CRC Discrete Probability Distributions 5.1 Bernoulli distribution 5.2 Beta binomial distribution 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 Beta Pascal distribution Binomial distribution Geometric distribution Hypergeometric distribution Multinomial distribution Negative binomial distribution Poisson distribution Rectangular (discrete uniform) distribution Continuous Probability Distributions 6.1 Arcsin distribution 6.2 Beta distribution 6.3 Cauchy distribution 6.4 6.5 6.6 6.7 Chi–square distribution Erlang distribution Exponential distribution Extreme–value distribution 6.8 6.9 6.10 F distribution Gamma distribution Half–normal distribution 6.11 6.12 6.13 Inverse Gaussian (Wald) distribution Laplace distribution Logistic distribution 6.14 6.15 6.16 Lognormal distribution Noncentral chi–square distribution Noncentral F distribution 6.17 6.18 6.19 Noncentral t distribution Normal distribution Normal distribution: multivariate 6.20 6.21 6.22 6.23 Pareto distribution Power function distribution Rayleigh distribution t distribution c 2000 by Chapman & Hall/CRC 6.24 6.25 6.26 6.27 Triangular distribution Uniform distribution Weibull distribution Relationships among distributions Standard Normal Distribution 7.1 Density function and related functions 7.2 Critical values 7.3 7.4 7.5 7.6 7.7 7.8 Tolerance factors for normal distributions Operating characteristic curves Multivariate normal distribution Distribution of the correlation coefficient Circular normal probabilities Circular error probabilities Estimation 8.1 Definitions 8.2 Cram´r–Rao inequality e 8.3 8.4 8.5 Theorems The method of moments The likelihood function 8.6 8.7 8.8 The method of maximum likelihood Invariance property of MLEs Different estimators 8.9 8.10 Estimators for small samples Estimators for large samples 9.1 9.2 9.3 9.4 9.5 9.6 9.7 10 Confidence Intervals Definitions Common critical values Sample size calculations Summary of common confidence intervals Confidence intervals: one sample Confidence intervals: two samples Finite population correction factor Hypothesis Testing c 2000 by Chapman & Hall/CRC 10.1 10.2 10.3 Introduction The Neyman–Pearson lemma Likelihood ratio tests 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 Goodness of fit test Contingency tables Bartlett’s test Cochran’s test Number of observations required Critical values for testing outliers Significance test in × contingency tables Determining values in Bernoulli trials 11 Regression Analysis 11.1 Simple linear regression 11.2 Multiple linear regression 11.3 Orthogonal polynomials 12 Analysis of Variance 12.1 12.2 12.3 12.4 12.5 12.6 13 One-way anova Two-way anova Three-factor experiments Manova Factor analysis Latin square design Experimental Design 13.1 Latin squares 13.2 Graeco–Latin squares 13.3 Block designs 13.4 13.5 13.6 13.7 Confounding in 2n factorial experiments Tables for design of experiments 13.8 14 Factorial experimentation: factors 2r Factorial experiments References Nonparametric Statistics 14.1 Friedman test for randomized block design c 2000 by Chapman & Hall/CRC 14.2 14.3 14.4 Kendall’s rank correlation coefficient Kolmogorov–Smirnoff tests Kruskal–Wallis test 14.5 14.6 14.7 14.8 14.9 14.10 The runs test The sign test Spearman’s rank correlation coefficient Wilcoxon matched-pairs signed-ranks test Wilcoxon rank–sum (Mann–Whitney) test Wilcoxon signed-rank test 15 Quality Control and Risk Analysis 15.1 Quality assurance 15.2 Acceptance sampling 15.3 Reliability 15.4 Risk analysis and decision rules 16 General Linear Models 16.1 Notation 16.2 16.3 16.4 16.5 16.6 17 The general linear model Summary of rules for matrix operations Quadratic forms General linear hypothesis of full rank General linear model of less than full rank Miscellaneous Topics 17.1 17.2 17.3 17.4 Geometric probability Information and communication theory Kalman filtering Large deviations (theory of rare events) 17.5 17.6 17.7 17.8 17.9 17.10 17.11 17.12 Markov chains Martingales Measure theoretical probability Monte Carlo integration techniques Queuing theory Random matrix eigenvalues Random number generation Resampling methods c 2000 by Chapman & Hall/CRC n 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 Γ(n) 1.0000 9943 9888 9835 9784 9735 9687 9642 9597 9555 9514 9474 9436 9399 9364 9330 9298 9267 9237 9209 9182 9156 9131 9108 9085 9064 n 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 Γ(n) 9064 9044 9025 9007 8990 8975 8960 8946 8934 8922 8912 8902 8893 8885 8879 8873 8868 8864 8860 8858 8857 8856 8856 8857 8859 8862 n 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 Γ(n) 8862 8866 8870 8876 8882 8889 8896 8905 8914 8924 8935 8947 8959 8972 8986 9001 9017 9033 9050 9068 9086 9106 9126 9147 9168 9191 n 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 Γ(n) 9191 9214 9238 9262 9288 9314 9341 9368 9397 9426 9456 9487 9518 9551 9584 9618 9652 9688 9724 9761 9799 9837 9877 9917 9958 1.0000 Table 18.5: Values of the gamma function 18.8.6 Incomplete gamma functions The incomplete gamma functions are defined by x γ(a, x) = e−t ta−1 dt Γ(a, x) = Γ(a) − γ(a, x) = ∞ x c 2000 by Chapman & Hall/CRC (18.48) e−t ta−1 dt 18.9 HYPERGEOMETRIC FUNCTIONS Recall the geometric series and the binomial expansion (|z| < 1): (1 − z)−1 = ∞ (1 − z)−a = zn, n=0 ∞ ∞ −a zn (a)n (−z)n = (18.49) n n! n=0 n=0 where Pochhammer’s symbol, (a)n , is defined in section 18.7 18.9.1 Generalized hypergeometric function The generalized hypergeometric function is defined by: p Fq a1 , a2 , , ap ; b1 , b2 , , bq ; x = p Fq ∞ = k=0 ∞ = k=0 a1 , , ap ; x b1 , , bq p i=1 (ai )k q i=1 (bi )k xk k! (18.50) (a1 )k · · · (ap )k xk (b1 )k · · · (bq )k k! where (n)k represents Pochhammer’s symbol Usually, F1 (a, b; c; x) is called “the” hypergeometric function; this is also called the Gauss hypergeometric function 18.9.2 Gauss hypergeometric function The Gauss hypergeometric function is ∞ F (a, b; c; x) = F1 (a, b; c; x) = 18.9.2.1 (a)n (b)n xn (c)n n! n=0 (18.51) Special cases F (a, b; b; x) = (1 − x)−a , ln(1 − x) , x 1+x ln , 2x 1−x arctan x , x arcsin x , x √ ln(x + + x2 ) x F (1, 1; 2; x) = − F ( 1/2, 1; 3/2; x2 ) = F ( 1/2, 1; 3/2; −x2 ) = F ( 1/2, 1/2; 3/2; x2 ) = F ( 1/2, 1/2; 3/2; −x2 ) = c 2000 by Chapman & Hall/CRC (18.52) 18.9.2.2 Functional relations F (a, b; c; x) = (1 − x)−a F = (1 − x)−b F x x−1 x c − a, b; c; x−1 a, c − b; c; (18.53) = (1 − x)c−a−b F (c − a, c − b; c; x) 18.9.3 Confluent hypergeometric functions The confluent hypergeometric functions, M and U , are defined by z M (a, c, z) = lim F a, b; c; z→∞ b (18.54) Γ(1 − c) Γ(c − 1) 1−c U (a, c, z) = M (a, c, z) + z M (a − c, − c, z) Γ(a − c + 1) Γ(a) Sometimes the notation ψ is used for M Sometimes U (a, b, z) is called the Tricomi function; it is the unique solution to zw + (b − z)w − aw = with U (a, b, 0) = Γ(1 − b)/Γ(1 + a − b) and U (a, b, ∞) = 18.10 LOGARITHMIC FUNCTIONS 18.10.1 Definition of the natural log The natural logarithm (also known as the Naperian logarithm) of z is written as ln z or as loge z It is sometimes written log z (this is also used to represent a “generic” logarithm, a logarithm to any base) One definition is z ln z = dt , t (18.55) where the integration path from to z does not cross the origin or the negative real axis For complex values of z the natural logarithm, as defined above, can be represented in terms of it’s magnitude and phase If z = x + iy = reiθ , then ln z = ln r + iθ, where r = x2 + y , x = r cos θ, and y = r sin θ 18.10.2 Special values lim (ln ) = −∞ →0 ln (−1) = iπ + 2πik c 2000 by Chapman & Hall/CRC ln = ln (±i) = ± ln e = iπ + 2πik 18.10.3 Logarithms to a base other than e The logarithmic function to the base a, written loga , is defined as loga z = logb z ln z = logb a ln a (18.56) Note the properties: (a) loga ap = p (b) loga b = logb a ln z (c) log10 z = = (log10 e) ln z ≈ (0.4342944819 ) ln z ln 10 (d) ln z = (ln 10) log10 z ≈ (2.3025850929 ) log10 z 18.10.4 Relation of the logarithm to the exponential For real values of z the logarithm is a monotonic function, as is the exponential Any monotonic function has a single–valued inverse function; the natural logarithm is the inverse of the exponential That is, if x = ey , then y = ln x and x = eln x The same inverse relations exist for bases other than e For example, if u = aw , then w = loga u and u = aloga u 18.10.5 Identities loga z1 z2 = loga z1 + loga z2 for (−π < arg z1 + arg z2 < π) z1 loga = loga z1 − loga z2 for (−π < arg z1 − arg z2 < π) z2 loga z n = n loga z for (−π < n arg z < π), when n is an integer 18.10.6 Series expansions for the natural logarithm 1 ln (1 + z) = z − z + z − · · · , z−1 z−1 ln z = + z z 18.10.7 for |z| < + z−1 z + · · · for Re(z) ≥ Derivative and integration formulae d ln z = dz z dz = ln |z| + C z c 2000 by Chapman & Hall/CRC ln z dz = z ln |z| − z + C (18.57) n p(n) n p(n) n p(n) n p(n) 11 15 22 30 11 12 13 14 15 16 17 56 77 101 135 176 231 297 21 22 23 24 25 26 792 1002 1255 1575 1958 2436 31 32 33 34 35 6842 8349 10143 12310 14883 10 42 18 19 20 385 490 627 27 28 29 30 3010 3718 4565 5604 40 45 50 37338 89134 204226 Table 18.6: Values of the partition function 18.11 PARTITIONS A partition of a number n is a representation of n as the sum of any number of positive integral parts (for example: = + = + = + + = + + = + + + = + + + + ) The number of partitions of n is p(n) (for example, p(5) = 7) The number of partitions of n into at most m parts is equal to the number of partitions of n into parts which not exceed m; this is denoted pm (n) (for example, p3 (5) = and p2 (5) = 3) The generating functions for p(n) and pm (n) are ∞ p(n)xn = 1+ n=1 ∞ n pm (n)xn tm = 1+ n=1 m=1 18.12 (1 − x)(1 − x2 )(1 − x3 ) · · · (18.58) (1 − tx)(1 − tx2 )(1 − tx3 ) · · · (18.59) SIGNUM FUNCTION The signum function indicates whether the argument is greater than or less than zero: sgn(x) = 18.13 −1 x>0 x 0, then n n n n x(n) = x+ x + ··· + x (18.62) n For example: x(3) = x(x − 1)(x − 2) = 2x − 3x2 + x3 = 3 x (3) Stirling numbers satisfy ∞ n=m n m xn n! = (log(1+x))m m! Example 18.86 : For the element set {a, b, c, d} there are containing exactly cycles They are: x+ x2 + for |x| < = 11 permutations 1234 2314 = (123)(4), 1234 3124 = (132)(4), 1234 3241 = (134)(2), 1234 4213 = (143)(2), 1234 2431 = (124)(3), 1234 4132 = (142)(3), 1234 1342 = (234)(1), 1234 1423 = (243)(1), 1234 2143 = (12)(34), 1234 3412 = (13)(24), 1234 4321 = (14)(23) c 2000 by Chapman & Hall/CRC n 10 11 12 13 14 15 m=1 1 1 1 1 1 1 1 7 15 31 63 127 255 511 1023 2047 4095 8191 16383 25 90 301 966 3025 9330 28501 86526 261625 788970 2375101 10 65 350 1701 7770 34105 145750 611501 2532530 10391745 42355950 15 140 1050 6951 42525 246730 1379400 7508501 40075035 210766920 21 266 2646 22827 179487 1323652 9321312 63436373 420693273 28 462 5880 63987 627396 5715424 49329280 408741333 Table 18.8: Table of Stirling cycle numbers 18.13.2 n m Stirling cycle numbers n The Stirling cycle number, m , is the number of ways to partition n into m blocks (Equivalently, it is the number of ways that n distinguishable balls can be placed in m indistinguishable cells, with no cell empty.) The Stirling cycle numbers can be numerically evaluated as (see Table 18.8): n m = m! m (−1)m−i i=0 m n i i (18.63) Ordinary powers can be expanded in terms of factorial polynomials If n > 0, then n n n xn = x(1) + x(2) + · · · + x(n) (18.64) n For example: x3 = x(1) + x(2) + x(3) Example 18.87 : Placing the distinguishable balls {a, b, c, d} into distinguishable cells, so that no cell is empty, can be done in delineate the cells): | ab | cd | | b | acd | 18.14 | ad | bc | | c | abd | = ways These are (vertical bars | ac | bd | | d | abc | | a | bcd | SUMS OF POWERS OF INTEGERS n Define sk (n) = 1k + 2k + · · · + nk = mk Properties include: m=1 (a) sk (n) = (k + 1)−1 [Bk+1 (n + 1) − Bk+1 (0)], where the Bk are Bernoulli polynomials c 2000 by Chapman & Hall/CRC n n n n k2 k k=1 n k3 k=1 k=1 n k4 k5 k=1 k=1 10 15 14 30 55 36 100 225 17 98 354 979 33 276 1300 4425 10 21 28 36 45 55 91 140 204 285 385 441 784 1296 2025 3025 2275 4676 8772 15333 25333 12201 29008 61776 120825 220825 11 12 13 14 15 66 78 91 105 120 506 650 819 1015 1240 4356 6084 8281 11025 14400 39974 60710 89271 127687 178312 381876 630708 1002001 1539825 2299200 16 17 18 19 20 136 153 171 190 210 1496 1785 2109 2470 2870 18496 23409 29241 36100 44100 243848 327369 432345 562666 722666 3347776 4767633 6657201 9133300 12333300 Table 18.9: Sums of powers of integers k+1 am nk−m+2 there is the recursion formula: (b) Writing sk (n) as m=1 sk+1 (n) = + ··· + k+1 k+2 k+1 a1 nk+2 + · · · + k+1 k a3 nk k+1 ak+1 n2 + − (k + 1) am n k+3−m m=1 n(n + 1) s2 (n) = 12 + 22 + 32 + · · · + n2 = n(n + 1)(2n + 1) 3 3 s3 (n) = + + + · · · + n = (n (n + 1)2 ) = [s1 (n)]2 4 4 s4 (n) = + + + · · · + n = (3n2 + 3n − 1)s2 (n) s1 (n) = + + + · · · + n = c 2000 by Chapman & Hall/CRC (18.65) s5 (n) = 15 + 25 + 35 + · · · + n5 = s6 (n) = s7 (n) = s8 (n) = s9 (n) = s10 (n) = 18.15 n (n + 1)2 (2n2 + 2n − 1) 12 n (n + 1)(2n + 1)(3n4 + 6n3 − 3n + 1) 42 n2 (n + 1)2 (3n4 + 6n3 − n2 − 4n + 2) 24 n (n + 1)(2n + 1)(5n6 + 15n5 + 5n4 − 15n3 − n2 + 9n − 3) 90 n2 (n + 1)2 (2n6 + 6n5 + n4 − 8n3 + n2 + 6n − 3) 20 n (n + 1)(2n + 1)(3n8 + 12n7 + 8n6 − 18n5 66 − 10n4 + 24n3 + 2n2 − 15n + 5) TABLES OF ORTHOGONAL POLYNOMIALS In the following: • • • • • Hn are Hermite polynomials Ln are Laguerre polynomials Pn are Legendre polynomials Tn are Chebyshev polynomials Un are Chebyshev polynomials x10 = (30240H0 + 75600H2 + 25200H4 + 2520H6 + 90H8 + H10 )/1024 H0 = x9 = (15120H1 + 10080H3 + 1512H5 + 72H7 + H9 )/512 H1 = 2x H2 = 4x2 − x8 = (1680H0 + 3360H2 + 840H4 + 56H6 + H8 )/256 H3 = 8x − 12x x7 = (840H1 + 420H3 + 42H5 + H7 )/128 H4 = 16x − 48x + 12 x6 = (120H0 + 180H2 + 30H4 + H6 )/64 H5 = 32x5 − 160x3 + 120x x5 = (60H1 + 20H3 + H5 )/32 H6 = 64x6 − 480x4 + 720x2 − 120 x4 = (12H0 + 12H2 + H4 )/16 H7 = 128x7 − 1344x5 + 3360x3 − 1680x x3 = (6H1 + H3 )/8 H8 = 256x − 3584x + 13440x − 13440x + 1680 x2 = (2H0 + H2 )/4 H9 = 512x9 − 9216x7 + 48384x5 − 80640x3 + 30240x x = (H1 )/2 H10 = 1024x10 − 23040x8 + 161280x6 − 403200x4 + 302400x2 − 30240 = H0 x6 = 720L0 − 4320L1 + 10800L2 − 14400L3 + 10800L4 − 4320L5 + 720L6 L0 = L1 = −x + x5 = 120L0 − 600L1 + 1200L2 − 1200L3 + 600L4 − 120L5 L2 = (x2 − 4x + 2)/2 x4 = 24L0 − 96L1 + 144L2 − 96L3 + 24L4 L3 = (−x + 9x − 18x + 6)/6 x3 = 6L0 − 18L1 + 18L2 − 6L3 L4 = (x − 16x + 72x − 96x + 24)/24 L5 = (−x5 + 25x4 − 200x3 + 600x2 − 600x + 120)/120 L6 = (x6 − 36x5 + 450x4 − 2400x3 + 5400x2 − 4320x + 720)/720 x2 = 2L0 − 4L1 + 2L2 x = L0 − L1 = L0 P0 = x10 = (4199P0 + 16150P2 + 15504P4 + 7904P6 + 2176P8 + 256P10 )/46189 P1 = x x9 = (3315P1 + 4760P3 + 2992P5 + 960P7 + 128P9 )/12155 c 2000 by Chapman & Hall/CRC P2 = (3x2 − 1)/2 x8 = (715P0 + 2600P2 + 2160P4 + 832P6 + 128P8 )/6435 P3 = (5x3 − 3x)/2 x7 = (143P1 + 182P3 + 88P5 + 16P7 )/429 P4 = (35x − 30x + 3)/8 x6 = (33P0 + 110P2 + 72P4 + 16P6 )/231 P5 = (63x − 70x + 15x)/8 x5 = (27P1 + 28P3 + 8P5 )/63 P6 = (231x6 − 315x4 + 105x2 − 5)/16 x4 = (7P0 + 20P2 + 8P4 )/35 P7 = (429x7 − 693x5 + 315x3 − 35x)/16 x3 = (3P1 + 2P3 )/5 P8 = (6435x − 12012x + 6930x − 1260x + 35)/128 x2 = (P0 + 2P2 )/3 P9 = (12155x − 25740x + 18018x − 4620x + 315x)/128 x = P1 P10 = (46189x10 − 109395x8 + 90090x6 − 30030x4 + 3465x2 − 63)/256 = P0 x10 = (126T0 + 210T2 + 120T4 + 45T6 + 10T8 + T10 )/512 T0 = x9 = (126T1 + 84T3 + 36T5 + 9T7 + T9 )/256 T1 = x T2 = 2x − x8 = (35T0 + 56T2 + 28T4 + 8T6 + T8 )/128 T3 = 4x3 − 3x x7 = (35T1 + 21T3 + 7T5 + T7 )/64 T4 = 8x4 − 8x2 + x6 = (10T0 + 15T2 + 6T4 + T6 )/32 T5 = 16x5 − 20x3 + 5x x5 = (10T1 + 5T3 + T5 )/16 T6 = 32x − 48x + 18x − x4 = (3T0 + 4T2 + T4 )/8 T7 = 64x7 − 112x5 + 56x3 − 7x x3 = (3T1 + T3 )/4 T8 = 128x8 − 256x6 + 160x4 − 32x2 + x2 = (T0 + T2 )/2 T9 = 256x − 576x + 432x − 120x + 9x x = T1 T10 = 512x10 − 1280x8 + 1120x6 − 400x4 + 50x2 − 1 = T0 x10 = (42U0 + 90U2 + 75U4 + 35U6 + 9U8 + U10 )/1024 U0 = U1 = 2x x9 = (42U1 + 48U3 + 27U5 + 8U7 + U9 )/512 U2 = 4x − x8 = (14U0 + 28U2 + 20U4 + 7U6 + U8 )/256 U3 = 8x3 − 4x x7 = (14U1 + 14U3 + 6U5 + U7 )/128 U4 = 16x4 − 12x2 + x6 = (5U0 + 9U2 + 5U4 + U6 )/64 U5 = 32x − 32x + 6x x5 = (5U1 + 4U3 + U5 )/32 U6 = 64x − 80x + 24x − x4 = (2U0 + 3U2 + U4 )/16 U7 = 128x7 − 192x5 + 80x3 − 8x x3 = (2U1 + U3 )/8 U8 = 256x8 − 448x6 + 240x4 − 40x2 + U9 = 512x − 1024x + 672x − 160x + 10x U10 = 1024x10 − 2304x8 + 1792x6 − 560x4 + 60x2 − 18.16 x2 = (U0 + U2 )/4 x = (U1 )/2 = U0 REFERENCES M Abramowitz and I A Stegun, Handbook of mathematical functions, NIST, Washington, DC, 1964, A Erd´lyi (ed.), Bateman Manuscript Project, Tables of integral transe forms, in volumes, McGraw–Hill, New York, 1954 N M Temme, Special functions: An introduction to the classical functions of mathematical physics, John Wiley & Sons, New York, 1996 c 2000 by Chapman & Hall/CRC List of Notation Symbols !!: double factorial !: factorial : complement of a set (1): treatment totals (a): treatment totals (ab): treatment totals (b): treatment totals (n)k : Pochhammer’s symbol n : binomial coefficient k n : multinomial coefficient n1 , ,nk : dot notation =: set equality T : transpose [A]: effect total for factor a : ceiling function : floor function n : Stirling numbers m { }: empty set n : Stirling cycle numbers m ¯: mean ∩: set intersection ∪: set union ∈: in set ∈: not in set |: conditional probability ⊕: exclusive or ∼: distribution similarity ⊂: subset ⊆: improper subset ⊃: superset ˜: median ˜: triangular matrix Greek Letters α: Weibull parameter α: confidence coefficient α: type I error (αβ)ij : level ij effect c 2000 by Chapman & Hall/CRC αi : ith treatment effect αi : level i factor A effect β: Weibull parameter β: type II error βj : level j factor B effect δi : Press residuals δ(x): delta function : error of estimation ij : error term ˆij : estimated expected count ijk : error term γ: Euler’s constant Γ(a, x): incomplete gamma function γ(a, x): incomplete gamma function Γ(x): gamma function κr : cumulant λ: parameter exponential distribution noncentral chi–square noncentrality of a BIBD Poisson distribution λ: test statistic λj : scaling factor µ: vector of means µ: parameter location noncentral chi–square scale µr : moment about the mean µ[r] : factorial moment µr : moment about the origin ν: parameter t distribution chi distribution chi–squared distribution shape ν1 : parameter F distribution shape ν2 : parameter F distribution shape φ: characteristic function φ: empty set ϕ(x): digamma function Φ(z): normal distribution function 166 ρij : correlation coefficient Σ: variance–covariance matrix σ: parameter Rayleigh distribution scale shape σ: standard deviation σ chart σ-field σ : variance σX|y : conditional variance σi : standard deviation σii : variance σij : covariance Σk/k−1 : error covariance matrix τ : Kendall’s Tau θ: distribution parameter θ: shape parameter ξi (x): orthogonal polynomials ξi (x): scaled orthogonal polynomials 276 Numbers 0: vector of all zeros 1: vector of all ones A A: interarrival time A: midrange a: location parameter A/B/c/K/m/Z: queue representation ALFS: additive lagged-Fibonacci sequence an : proportion of customers anova: analysis of variance AOQ: average outgoing quality c 2000 by Chapman & Hall/CRC AOQL: average outgoing quality limit AQL: acceptable quality level AR(k): autoregressive model ARMA(k, l): mixed model B B: service time B: vector of block totals b: parameter of a BIBD b: power function parameter B(a, b): beta function B[ ]: bias BIBD: balanced incomplete block design C C: channel capacity b: power function parameter c: bin width c: number of identical servers c chart C(n, k): k-combination C(n; n1 , , nk ): multinomial coefficient cdf: cumulative distribution function 33 CF: cumulative frequency ch: characteristic roots cosh(x): hyperbolic function cos(x): circular function CQV: coefficient of quartile variation 17 C R (n, k): k-combination with replacement c(t): cumulant generating function 38 CV: coefficient of variation D D: Kolmogorov–Smirnoff statistic 346, 348 D: constant service time D+ : Kolmogorov–Smirnoff statistic 346, 348 δij : Kronecker delta det(X): determinant of matrix X 404 Di : Cook’s distance Dn : derangement dn : proportion of customers Dx : diagonal matrix E eij : observed value E[ ]: expectation ei : residual Ek : Erlang-k service time erf: error function erfc: complementary error function 512 F F : Fourier transform FCFS: first come, first served FIFO: first in, first out fk : frequency F (x): cumulative distribution function F (x1 , x2 , , xn ): cumulative distribution function f (x1 , x2 , , xn ): probability density function f (x | y): conditional probability G G: general service time distribution 441 g1 : coefficient of skewness g2 : coefficient of skewness GI: general interarrival time GM: geometric mean IQR: interquartile range J J: determinant of the Jacobian jn (z): half order Bessel function Jν (z): Bessel function K K: system capacity k: parameter of a BIBD Kk : Kalman gain matrix Kν (z): Bessel function 509 KX (t1 , t2 ): correlation function L L: average number of customers L(θ): expected loss function L(θ): likelihood function (θ, a): loss function λ: average arrival rate λlower : confidence interval λupper : confidence interval LCG: linear congruential generator 450 LCL: lower control limit LIFO: last in, first out Ln (x): Laguerre polynomial ln: logarithm log: logarithm logb : logarithm to base b Lq : average number of customers LTPD: lot tolerance percent defective H M H(pX ): entropy H0 : null hypothesis Ha : alternative hypothesis Hk : k-stage hyperexponential service time HM: harmonic mean Hn (x): Hermite polynomial M : exponential service time M : hypergeometric function m: number in the source MA(l): moving average M/D/1: queue MD: mean deviation M/Ek /1: queue mgf: moment generating function MLE: maximum likelihood estimator I I: identity matrix I(X, Y ): mutual information iid: independent and identically distributed Iν (z): Bessel function c 2000 by Chapman & Hall/CRC M/M/1: queue M/M/1/K: queue M/M/2: queue M/M/c: queue M/M/c/c: queue M/M/c/K: queue M/M/∞: queue Mo : mode mr : moment about the mean mr : moment about the origin ms-lim: mean square limit MSE: mean square—error MSR: mean square—regression MTBF: mean time between failures PRNG: pseudorandom number generator Prob[ ]: probability P (t): factorial moment generating function p(x): probability mass function pX×Y : joint probability distribution p(x | y): conditional probability Q µ: average service rate µp : MTBF for parallel system µs : MTBF for series system MVUE: minimum variance unbiased estimator mX (t): moment generating function 37 µX|y : conditional mean N N : natural numbers n: shape parameter ne : degrees of freedom—errors nh : degrees of freedom—hypothesis 410 P p chart p(n) partitions P (n, k): k-permutation P (x, y): Markov transition function p-value ∂()/∂(): derivative pdf: probability density function π(x): probability distribution pm (n) restricted partitions pmf: probability mass function pn : proportion of time Pn (x): Legendre polynomial P n (x, y): n-step Markov transition matrix P R (n, k): k-permutation with replacement Per(xn ): period of a sequence PRESS: prediction sum of squares PRI: priority service c 2000 by Chapman & Hall/CRC QD: quartile deviation Qi : ith quartile R R: range R: rate of a code R: real numbers r: parameter of a BIBD r: sample correlation coefficient R chart R(t): reliability function r(θ, a): regret function R(θ, di ): risk function R2 : coefficient multiple determination Ra : adjusted coefficient multiple determination Re: real part ρ: server utilization Ri : reliability of a component RMS: root mean square Rp : reliability of parallel system RR: rejection region Rs : reliability of series system rS : Spearman’s rank correlation coefficient rS,α : Spearman’s rank correlation coefficient RSS: random service S S: sample space S: universal set s: sample standard deviation S : mean square—error S : sample variance s2 sample variance SC : mean square—columns SEM: standard error—mean sgn: signum function sinh(x): hyperbolic function sin(x): circular function Sk1 : coefficient of skewness Sk2 : coefficient of skewness Skq : coefficient of skewness SPRT: sequential probability ratio test SR mean square—rows SR : mean square—rows SRS: shift register sequence SSA: sum of squares—treatment SSE(F): sum of squares—error—full model SSE(F): sum of squares—error—reduced model SSE: sum of squares—error 263, 269, 283, 415 SSH: sum of squares—hypothesis 415 SSLF: sum of squares—lack of fit 266 SSPE: sum of squares—pure error 266 SSR: sum of squares—regression 263, 269 SST: sum of squares—total 263, 269, 283 St : Kendall’s score STr : mean square—treatments Sxx Sxy Syy T T : sample total T: vector of treatment totals T : sum of all observations Ti : sum of ith observations Tn (x): Chebyshev polynomial tr: trace of a matrix TS: test statistic tx,y transition probabilities U U : Mann–Whitney U statistic U : hypergeometric function u: traffic intensity c 2000 by Chapman & Hall/CRC UCL: upper control limit ui : coded class mark UMV: uniformly minimum variance unbiased Un (x): Chebyshev polynomial V v: parameter of a BIBD vk : Gaussian white process VR: variance ratio W W : average time W : range, standardized W : range, studentized W (t): Brownian motion wi : weight wk : Gaussian white process Wq : average time X X: design matrix X: random vector x: column vector X(i) : order statistic X: sample mean x: mean of x x chart xi : class mark xk/k−1 : estimate of xk xo : computing origin x: median of x ˜ xtr(p) : trimmed mean of x Y y : mean of all observations y i : mean of ith observation yij : observed value of Yij yn (z): half order Bessel function Yν (z): Bessel function Z Z: queue discipline Z(t): instantaneous failure rate Zk−1 : sequence of observed values ... HALL /CRC Boca Raton London New York Washington, D.C Library of Congress Cataloging-in-Publication Data Zwillinger, Daniel, 195 7CRC standard probability and statistics tables and formulae / Daniel. . .CRC standard probability and Statistics tables and formulae DANIEL ZWILLINGER Rensselaer Polytechnic Institute Troy, New York STEPHEN... i=1 Standard deviation The standard deviation is the positive square root of the variance: s = √ s2 The probable error is 0.6745 times the standard deviation 2.2.14 Standard errors The standard