Products of Random Variables

  • John H. Drew
  • Diane L. Evans
  • Andrew G. Glen
  • Lawrence M. Leemis
Chapter
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 246)

Abstract

This chapter describes an algorithm for computing the PDF of the product of two independent continuous random variables. This algorithm has been implemented in the Product procedure in APPL. The algorithms behind the Transform and BiTransform procedures from the two previous chapters differ fundamentally from the algorithm behind the Product procedure in that the transformation algorithms are more general whereas determining the distribution of the product of two random variables is more specific. Some examples given in the chapter demonstrate the algorithm’s application.

References

  1. 1.
    Abate J, Whitt W (1988) Transient behavior of the MM∕1 queue via Laplace transforms. Adv Appl Probab 20:145–178CrossRefGoogle Scholar
  2. 2.
    Adlakha VG, Kulkarni VG (1989) A classified bibliography of research on stochastic PERT networks: 1966–1987. INFOR 27:272–296Google Scholar
  3. 3.
    Andrews DWK, Buchinsky M (2000) A three-step method for choosing the number of bootstrap repetitions. Econometrica 68:23–51CrossRefGoogle Scholar
  4. 4.
    Andrews DWK, Buchinsky M (2002) On the number of bootstrap repetitions for BCa confidence intervals. Econometric Theory 18:962–984Google Scholar
  5. 5.
    Arnold BC, Balakrishnan N, Nagaraja HN (1992) A first course in order statistics. SIAM, PhiladelphiaGoogle Scholar
  6. 6.
    Balakrishnan N, Chen WWS (1997) CRC handbook of tables for order statistics from inverse Gaussian distributions with applications. CRC Press, Boca RatonGoogle Scholar
  7. 7.
    Banks J, Carson JS, Nelson BL, Nicol DM (2005) Discrete-event system simulation, 4th edn. Prentice-Hall, Upper Saddle River, New JerseyGoogle Scholar
  8. 8.
    Barr D, Zehna PW (1971) Probability. Brooks/ColeGoogle Scholar
  9. 9.
    Benford F (1938) The law of anomalous numbers. Proc Am Philos Soc 78: 551–572Google Scholar
  10. 10.
    Berger A, Hill, TP (2015) An introduction to Benford’s law. Princeton University Press, PrincetonCrossRefGoogle Scholar
  11. 11.
    Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New YorkGoogle Scholar
  12. 12.
    Birnbaum ZW (1952) Numerical tabulation of the distribution of Kolomogorov’s statistic for finite sample size. J Am Stat Assoc 47:425–441CrossRefGoogle Scholar
  13. 13.
    Box GEP, Jenkins GM (1994) Time series analysis: forecasting & control, 3rd edn. Prentice-HallGoogle Scholar
  14. 14.
    Burr IW (1955) Calculation of exact sampling distribution of ranges from a discrete population. Ann Math Stat 26:530–532 (correction, volume 38, 280)Google Scholar
  15. 15.
    Carrano FM, Helman P, Veroff R (1998) Data abstraction and problem solving with C++: walls and mirrors, 2nd edn. Addison-Wesley Longman, Reading, MassachusettsGoogle Scholar
  16. 16.
    Casella G, Berger R (2002) Statistical inference, 2nd edn. Duxbury, Pacific Grove, CaliforniaGoogle Scholar
  17. 17.
    Ciardo G, Leemis LM, Nicol D (1995) On the minimum of independent geometrically distributed random variables. Stat Probab Lett 23:313–326CrossRefGoogle Scholar
  18. 18.
    Cook P, Broemeling LD (1995) Bayesian statistics using mathematica. Am Stat 49:70–76Google Scholar
  19. 19.
    D’Agostino RB, Stephens MA (1986) Goodness-of-fit techniques. Marcel Dekker, New YorkGoogle Scholar
  20. 20.
    David HA, Nagaraja HN (2003) Order statistics, 3rd edn. Wiley, New YorkCrossRefGoogle Scholar
  21. 21.
    Devroye L (1996) Random variate generation in one line of code. In: Charnes J, Morrice D, Brunner D, Swain J (eds) Proceedings of the 1996 winter simulation conference. Institute of Electrical and Electronics Engineers, Coronado, CA, pp 265–272Google Scholar
  22. 22.
    Doss H, Chiang Y (1994) Choosing the resampling scheme when bootstrapping: a case study in reliability. J Am Stat Assoc 89:298–308CrossRefGoogle Scholar
  23. 23.
    Drew JH, Glen AG, Leemis LM (2000) Computing the cumulative distribution function of the Kolmogorov–Smirnov statistic. Comput Stat Data Anal 34: 1–15CrossRefGoogle Scholar
  24. 24.
    Duggan MJ, Drew JH, Leemis LM (2005) A test of randomness based on the distance between consecutive random number pairs. In: Kuhl ME, Steiger NM, Armstrong FB, Joines JA (eds) Proceedings of the 2005 winter simulation conference. Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, pp 741–748CrossRefGoogle Scholar
  25. 25.
    e Silva ES, Gail HR, Campos RV (1995) Calculating transient distributions of cumulative reward. In: Proceedings of the 1995 ACM SIGMETRICS joint international conference on measurement and modeling of computer systems, pp 231–240Google Scholar
  26. 26.
    Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman & Hall, New YorkCrossRefGoogle Scholar
  27. 27.
    Elmaghraby SE (1977) Activity networks: project planning and control by network models. Wiley, New YorkGoogle Scholar
  28. 28.
    Evans DL, Leemis LM (2000) Input modeling using a computer algebra system. In: Joines J, Barton R, Fishwick P, Kang K. (eds) Proceedings of the 2000 winter simulation conference. Institute of Electrical and Electronics Engineers, Piscataway, New Jersey, pp 577–586CrossRefGoogle Scholar
  29. 29.
    Evans DL, Leemis LM (2004) Algorithms for determining the distributions of sums of discrete random variables. Math Comput Modell 40:1429–1452CrossRefGoogle Scholar
  30. 30.
    Evans DL, Leemis LM, Drew JH (2006) The distribution of order statistics for discrete random variables with applications to bootstrapping. INFORMS J Comput 18:19–30CrossRefGoogle Scholar
  31. 31.
    Evans DL, Drew JH, Leemis, LM (2008) The distribution of the Kolmogorov–Smirnov, Cramer–von Mises, and Anderson–Darling test statistics for exponential populations with estimated parameters. Commun Stat–Simul Comput 37:1396–1421CrossRefGoogle Scholar
  32. 32.
    Fisher DL, Saisi D, Goldstein WM (1985) Stochastic PERT networks: OP diagrams, critical paths and the project completion time. Comput Oper Res 12:471–482CrossRefGoogle Scholar
  33. 33.
    Fishman GS (2001) Discrete-event simulation: modeling, programming, and analysis. Springer, BerlinCrossRefGoogle Scholar
  34. 34.
    Gafarian AV, Ancker CJ Jr, Morisaku T (1976) The problem of the initial transient in digital computer simulation. In: Proceedings of the 76 bicentennial conference on winter simulation, pp 49–51Google Scholar
  35. 35.
    Gehan EA (1965) A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika 52(parts 1 and 2):203–223Google Scholar
  36. 36.
    Ghosal S, Ghosh JK, Ramamoorthi RV (1999) Consistency issues in Bayesian nonparametrics. In: Ghosh S (ed) Asymptotics, nonparametrics and time series: a tribute to Madan Lal Puri. Statistics textbooks and monographs, vol 158. Marcel Dekker, New York, pp 639–667Google Scholar
  37. 37.
    Glen AG, Drew JH, Leemis LM (1997) A generalized univariate change-of-variable transformation technique. INFORMS J Comput 9:288–295CrossRefGoogle Scholar
  38. 38.
    Glen AG, Evans DL, Leemis LM (2001) APPL: a probability programming language. Am Stat 55:156–166CrossRefGoogle Scholar
  39. 39.
    Glen AG, Leemis LM, Drew JH (2004) Computing the distribution of the product of two continuous random variables. Comput Stat Data Anal 44: 451–464CrossRefGoogle Scholar
  40. 40.
    Grassmann WK (1977) Transient solutions in Markovian queueing systems. Comput Oper Res 4:47–53CrossRefGoogle Scholar
  41. 41.
    Grassmann WK (2008) Warm-up periods in simulation can be detrimental. Probab Eng Inform Sci 22:415–429CrossRefGoogle Scholar
  42. 42.
    Grinstead CM, Snell JL (1997) Introduction to probability, 2nd rev. edn. American Mathematical Society, Providence, Rhode IslandGoogle Scholar
  43. 43.
    Hagwood C (2009) An application of the residue calculus: the distribution of the sum of nonhomogeneous gamma variates. Am Stat 63:37–39CrossRefGoogle Scholar
  44. 44.
    Hamilton JD (1994) Time series analysis. Princeton University Press, PrincetonGoogle Scholar
  45. 45.
    Harter HL, Balakrishnan N (1996) CRC handbook of tables for the use of order statistics in estimation. CRC Press, Boca RatonGoogle Scholar
  46. 46.
    Hasting KJ (2006) Introduction to the mathematics of operations research with mathematica, 2nd edn. CRC Press, Boca RatonGoogle Scholar
  47. 47.
    Hill TP (1995) A statistical derivation of the significant-digit law. Stat Sci 86:354–363Google Scholar
  48. 48.
    Hill TP (1998) The first digit phenomenon. Am Sci 86:358–363CrossRefGoogle Scholar
  49. 49.
    Hillier FS, Lieberman GJ (2010) Introduction to operations research, 9th edn. McGraw–Hill, New YorkGoogle Scholar
  50. 50.
    Hogg RV, Craig AT (1995) Introduction to the mathematical statistics, 5th edn. Prentice-Hall, Upper Saddle River, New JerseyGoogle Scholar
  51. 51.
    Hogg RV, Tanis EA (2001) Probability and statistical inference, 6th edn. Prentice-Hall, Upper Saddle River, New JerseyGoogle Scholar
  52. 52.
    Hogg RV, McKean JW, Craig AT (2005) Introduction to the mathematical statistics, 6th edn. Prentice-Hall, Upper Saddle River, New JerseyGoogle Scholar
  53. 53.
    Hutson AD, Ernst MD (2000) The exact bootstrap mean and variance of an L-estimator. J R Stat Soc Ser B 62:89–94CrossRefGoogle Scholar
  54. 54.
    Jaakkola TS, Jordan MI (2000) Bayesian parameter estimation via variational methods. Stat Comput 10:25–37CrossRefGoogle Scholar
  55. 55.
    Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2, 2nd ed. Wiley, New YorkGoogle Scholar
  56. 56.
    Kaczynski WH, Leemis LM, Drew JH (2012) Transient queueing analysis. INFORMS J Comput 24:10–28CrossRefGoogle Scholar
  57. 57.
    Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, Hoboken, New JerseyCrossRefGoogle Scholar
  58. 58.
    Karian ZA, Tanis EA (1999) Probability and statistics: explorations with Maple, 2nd edn. Prentice-Hall, Upper Saddle River, New JerseyGoogle Scholar
  59. 59.
    Kelton WD (1985) Transient exponential-Erlang queues and steady-state simulation. Commun ACM 28:741–749CrossRefGoogle Scholar
  60. 60.
    Kelton WD, Law AM (1985) The transient behavior of the MMs queue, with implications for steady-state simulation. Oper Res 33:378–396CrossRefGoogle Scholar
  61. 61.
    Khoshkenara A, Mahloojia H. (2013) A new test of randomness for Lehmer generators based on the Manhattan Distance Between Pairs. Commun Stat–Simul Comput 42:202–214CrossRefGoogle Scholar
  62. 62.
    Kleinrock L (1975) Queueing systems. Wiley, New YorkGoogle Scholar
  63. 63.
    Knuth DE (1998) The art of computer programming, volume 2: seminumerical algorithms, 3rd edn. Addison-Wesley, Reading, MassachusettsGoogle Scholar
  64. 64.
    Kossovsky AE (2015) Benford’s law: theory, the general law of relative quantities, and forensic fraud detection applications. World Scientific, SingaporeGoogle Scholar
  65. 65.
    L’Écuyer P, Cordeau J-F, Simard R (2000) Close-point spatial tests and their application to random number generators. Oper Res 48:308–317CrossRefGoogle Scholar
  66. 66.
    Law AM (1975) A comparison of two techniques for determining the accuracy of simulation output. Technical Report 75–11, University of Wisconsin at MadisonGoogle Scholar
  67. 67.
    Laplante PA (ed) (2001) Dictionary of computer science, engineering and technology. CRC PressGoogle Scholar
  68. 68.
    Larsen RJ, Marx ML (2001) An introduction to mathematical statistics and its applications, 3rd edn. Prentice-Hall, Upper Saddle River, New JerseyGoogle Scholar
  69. 69.
    Larsen RJ, Marx ML (2006) An introduction to mathematical statistics and its applications, 4th edn. Prentice-Hall, Upper Saddle River, New JerseyGoogle Scholar
  70. 70.
    Law AM (2015) Simulation modeling and analysis, 5th edn. McGraw-Hill, New YorkGoogle Scholar
  71. 71.
    Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, Hoboken, New JerseyGoogle Scholar
  72. 72.
    Leemis L (1995) Reliability: probabilistic models and statistical methods. Prentice-Hall, Upper Saddle River, New JerseyGoogle Scholar
  73. 73.
    Leemis L (2006) Lower system reliability bounds from binary failure data using bootstrapping. J Qual Technol 38:2–13Google Scholar
  74. 74.
    Leemis L, Schmeiser B, Evans D (2000) Survival distributions satisfying Benford’s law. Am Stat 54:236–241Google Scholar
  75. 75.
    Leemis LM, Duggan MJ, Drew JH, Mallozzi JA, Connell KW (2006) Algorithms to calculate the distribution of the longest path length of a stochastic activity network with continuous activity durations. Networks 48:143–165CrossRefGoogle Scholar
  76. 76.
    Leguesdron P, Pellaumail J, Rubino G, Sericola B (1993) Transient analysis of the MM∕1 queue. Adv Appl Probab 25:702–713CrossRefGoogle Scholar
  77. 77.
    Lehmer DH (1951) Mathematical methods in large-scale computing units. In: Proceedings of the 2nd symposium on large-scale calculating machinery. Harvard University Press, pp 141–146Google Scholar
  78. 78.
    Ley E (1996) On the peculiar distribution of the U.S. stock indices digits. Am Stat 50:311–313Google Scholar
  79. 79.
    Lieblein J, Zelen M (1956) Statistical investigation of the fatigue life of deep-groove ball bearings. J Res Natl Bur Stand 57:273–316CrossRefGoogle Scholar
  80. 80.
    Maplesoft (2013) Maple, Version 17. WaterlooGoogle Scholar
  81. 81.
    Margolin BH, Winokur HS (1967) Exact moments of the order statistics of the geometric distribution and their relation to inverse sampling and reliability of redundant systems. J Am Stat Assoc 62:915–925CrossRefGoogle Scholar
  82. 82.
    Marks CE, Glen AG, Robinson MW, Leemis LM (2014) Applying bootstrap methods to system reliability. Am Stat 68:174–182CrossRefGoogle Scholar
  83. 83.
    Marsaglia G (1968) Random numbers fall mainly in the planes. Proc Natl Acad Sci 61:25–28CrossRefGoogle Scholar
  84. 84.
    Martin JJ (1965) Distribution of the time through a directed, acyclic network. Oper Res 13:44–66CrossRefGoogle Scholar
  85. 85.
    Martin MA (1990) On bootstrap iteration for coverage correction in confidence intervals. J Am Stat Assoc 85:1105–1118CrossRefGoogle Scholar
  86. 86.
    Meeker WQ, Escobar LA (1998) Statistical methods for reliability data. Wiley, New YorkGoogle Scholar
  87. 87.
    Miller LH (1956) Table of percentage points of Kolmogorov statistics. J Am Stat Assoc 51:111–121CrossRefGoogle Scholar
  88. 88.
    Miller S (ed) (2015) Benford’s law: theory & applications. Princeton University Press, PrincetonGoogle Scholar
  89. 89.
    Miller I, Miller M (2004) John E. Freund’s mathematical statistics, 7th edn. Prentice-Hall, Upper Saddle River, New JerseyGoogle Scholar
  90. 90.
    Morisaku T (1976) Techniques for data-truncation in digital computer simulation. Ph.D. thesis, University of Southern California, Los AngelesGoogle Scholar
  91. 91.
    Nelson BL, Yamnitsky M (1998) Input modeling tools for complex problems. In: Medeiros DJ, Watson EF, Carson JS, Manivannan, MS (eds) Proceedings of the 1998 winter simulation conference. Institute of Electrical and Electronics Engineers, pp 105–112Google Scholar
  92. 92.
    Nicol D (2000) Personal communicationGoogle Scholar
  93. 93.
    Nigrini M (1996) A taxpayer compliance application of Benford’s law. J Am Taxat Assoc 18:72–91Google Scholar
  94. 94.
    Odoni AR, Roth E (1983) Empirical investigation of the transient behavior of stationary queueing systems. Oper Res 31:432–455CrossRefGoogle Scholar
  95. 95.
    Owen DB (1962) Handbook of statistical tables. Addison-Wesley, Reading, MassachusettsGoogle Scholar
  96. 96.
    Padgett WJ, Tomlinson MA (2003) Lower confidence bounds for percentiles of Weibull and Birnbaum–Saunders distributions. J Stat Comput Simul 73: 429–443CrossRefGoogle Scholar
  97. 97.
    Park SK, Miller KW (1988) Random number generators: good ones are hard to find. Commun ACM 31:1192–1201CrossRefGoogle Scholar
  98. 98.
    Parlar M (2000) Interactive operations research with Maple. Birkhäuser, BostonCrossRefGoogle Scholar
  99. 99.
    Parthasarathy PR (1987) A transient solution to an MM∕1 queue: a simple approach. Adv Appl Probab 19:997–998CrossRefGoogle Scholar
  100. 100.
    Parzen E (1960) Modern probability theory and its applications. Wiley, New YorkGoogle Scholar
  101. 101.
    Pegden CD, Rosenshine M (1982) Some new results for the MM∕1 queue. Manage Sci 28:821–828CrossRefGoogle Scholar
  102. 102.
    Port SC (1994) Theoretical probability for applications. WileyGoogle Scholar
  103. 103.
    Rice JA (2007) Mathematical statistics and data analysis, 3rd edn. Thompson and Brooks/Cole, Belmont, CaliforniaGoogle Scholar
  104. 104.
    Rohatgi VK (1976) An introduction to probability theory and mathematical statistics. Wiley, New YorkGoogle Scholar
  105. 105.
    Rose C, Smith MD (2002) Mathematical statistics and mathematica. Springer, New YorkCrossRefGoogle Scholar
  106. 106.
    Ross S (2006) A first course in probability, 7th edn. Prentice Hall, Upper Saddle River, New JerseyGoogle Scholar
  107. 107.
    Ruskey F, Williams A (2008) Generating balanced parentheses and binary trees by prefix shifts. In: Proceedings of the 12th computing: the Australasian theory symposium (CATS2008), CRPIT, vol 77, pp 107–115Google Scholar
  108. 108.
    Shier DR (1991) Network reliability and algebraic structures. Oxford University Press, New YorkGoogle Scholar
  109. 109.
    Springer MD (1979) The algebra of random variables. Wiley, New YorkGoogle Scholar
  110. 110.
    Srivastava RC (1974) Two characterizations of the geometric distribution. J Am Stat Assoc 69:267–269CrossRefGoogle Scholar
  111. 111.
    Stanley RP (1999) Enumerative combinatorics. Volume 62 of Cambridge studies in advanced mathematics. Cambridge University Press, CambridgeGoogle Scholar
  112. 112.
    Thompson P (2000) Getting normal probability approximations without using normal tables. College Math J 31:51–54CrossRefGoogle Scholar
  113. 113.
    Trosset M (2001) Personal communicationGoogle Scholar
  114. 114.
    Vargo E, Pasupathy R, Leemis L (2010) Moment-ratio diagrams for univariate distributions. J Qual Technol 42:276–286Google Scholar
  115. 115.
    Webb KH, Leemis LM (2014) Symbolic ARMA model analysis. Comput Econ 43:313–330CrossRefGoogle Scholar
  116. 116.
    Weiss MA (1994) Data structures and algorithm analysis in C++. Addison-Wesley Publishing Company, Menlo Park, CaliforniaGoogle Scholar
  117. 117.
    Weisstein EW (2016) “Generalized Hypergeometric Function” from MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GeneralizedHypergeometricFunction.html
  118. 118.
    Winston WL (2004) Operations research: applications and algorithms, 4th edn. Thompson, Belmont, CaliforniaGoogle Scholar
  119. 119.
    Wolfram Research, Inc. (2013) Mathematica, Version 10, Champaign, ILGoogle Scholar
  120. 120.
    Woodward WA, Gray HL (1981) On the relationship between the S array and the Box–Jenkins method of ARMA model identification. J Am Stat Assoc 76:579–587CrossRefGoogle Scholar
  121. 121.
    Woodward JA, Palmer CGS (1997) On the exact convolution of discrete random variables. Appl Math Comput 83:69–77Google Scholar
  122. 122.
    Yang JX, Drew JH, Leemis LM (2012) Automating bivariate transformations. INFORMS J Comput 24:1–9CrossRefGoogle Scholar
  123. 123.
    Young DH (1970) The order statistics of the negative binomial distribution. Biometrika 57:181–186CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • John H. Drew
    • 1
  • Diane L. Evans
    • 2
  • Andrew G. Glen
    • 3
  • Lawrence M. Leemis
    • 1
  1. 1.Department of MathematicsThe College of William and MaryWilliamsburgUSA
  2. 2.Department of MathematicsRose-Hulman Institute of TechnologyTerre HauteUSA
  3. 3.Department of Mathematics and Computer ScienceColorado CollegeColorado SpringsUSA

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