Sampling from Probability Distributions

  • George S. Fishman
Part of the Springer Series in Operations Research book series (ORFE)


Virtually every commercially available product for performing discrete-event simulation incorporates software for sampling from diverse probability distributions. Often, this incorporation is relatively seamless, requiring the user merely to pull down a menu of options, select a distribution, and specify its parameters. This major convenience relieves the user of the need to write her or his code to effect sampling.


Rejection Method Poisson Generation Precomputed Table Fast Rejection Binomial Generation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Abramowitz, M., and I. Stegun (1964). Handbook of Mathematical Functions, Applied Mathematics Series 55, National Bureau of Standards, Washington, D.C.MATHGoogle Scholar
  2. Afflerbach, L., and W. Hörmann (1990). Nonuniform random numbers: a sensitivity analysis for tranformation methods, Lecture Notes in Economics and Mathematical Systems, G. Plug and U. Dieter, eds., Springer-Verlag, New York.Google Scholar
  3. Agrawal, A., and A. Satyanarayana (1984). An O (le I) time algorithm for computing the reliability of a class of directed networks, Oper. Res., 32, 493–515.MathSciNetMATHCrossRefGoogle Scholar
  4. Aho, A.V., J.E. Hoperoft, and J.D. Ullman (1974). The Design and Analysis of Computer Algorithms, Addison-Wesley, Reading, MA.Google Scholar
  5. Ahrens, J.H. (1989). How to avoid logarithms in comparisons with uniform random variables, Computing, 41, 163–166.MathSciNetMATHCrossRefGoogle Scholar
  6. Ahrens, J.H. (1993). Sampling from general distributions by suboptimal division of domains, Grazer Math. Berichte 319, Graz, Austria.Google Scholar
  7. Ahrens, J.H., and U. Dieter (1972). Computer methods for sampling from the exponential and normal distributions, Comm. ACM, 15, 873–882.MathSciNetMATHCrossRefGoogle Scholar
  8. Ahrens, J.H., and U. Dieter (1974a). Non-Uniform Random-Numbers, Institut für Math. Statistik, Technische Hochschule in Graz, Austria.Google Scholar
  9. Ahrens, J.H., and U. Dieter (1974b). Computer methods for sampling from Gamma Beta, Poisson and binomial distributions, Computing, 12, 223–246.MathSciNetMATHCrossRefGoogle Scholar
  10. Ahrens, J.H., and U. Dieter (1980a). Sampling from the binomial and Poisson distributions: a method with bounded computation times, Computing, 25, 193–208.MathSciNetMATHCrossRefGoogle Scholar
  11. Ahrens, J.H., and U. Dieter (1982a). Generating Gamma variates by a modified rejection technique, Comm. ACM, 25, 47–53.MathSciNetMATHCrossRefGoogle Scholar
  12. Ahrens, J.H., and U. Dieter (1982b). Computer generation of Poisson deviates from modified normal distributions, ACM Trans. Math. Software, 8, 163–179.MathSciNetMATHCrossRefGoogle Scholar
  13. Ahrens, J.H., and U. Dieter (1985). Sequential random sampling, ACM Transactions on Mathematical Software, 11, 157–169.MATHCrossRefGoogle Scholar
  14. Ahrens, J.H., and U. Dieter (1988). Efficient tablefree sampling methods for the exponential, Cauchy and normal distributions, Comm. ACM, 31, 1330–1337.CrossRefGoogle Scholar
  15. Ahrens, J.H., and U. Dieter (1991). A convenient sampling method with bounded computation times for Poisson distributions, The Frontiers of Statistical Computation, Simulation and Modeling. P.R. Nelson, E.J. Dudewicz, A.Öztürk, and E.C. van der Meulen, editors American Science Press, Syracuse, NY, 137–149.Google Scholar
  16. Anderson, T.W. (1958). An Introduction to Multivariate Statistical Analysis, John Wiley and Sons.Google Scholar
  17. Atkinson, A.C. (1979). The computer generation of Poisson random variables, Appl. Statist. 28, 29–35.MATHCrossRefGoogle Scholar
  18. Atkinson, A.C., and J. Whittaker (1976). A switching algorithm for the generation of beta random variables with at least one parameter less than one, J. Roy. Statist. Soc., Series A, 139, 462–467.MathSciNetCrossRefGoogle Scholar
  19. Best, D.J. (1978). A simple algorithm for the computer generation of random samples from a Student’s t or symmetric beta distribution, Proceedings of the 1978 COMPSTAT Conference, Leiden, 341–347.Google Scholar
  20. Best, D.J. (1983). A note on gamma variate generators with shape parameter less than unity, Computing, 30, 185–188.MathSciNetMATHCrossRefGoogle Scholar
  21. Box, G.E.P., and M.E. Muller (1958). A note on the generation of random normal deviates, Ann. Math. Statist, 29, 610–611.MATHCrossRefGoogle Scholar
  22. Broder, A.Z. (1989). Generating random spanning trees, Thirtieth Annual Symposium on Foundations of Computer Science, 442–447.Google Scholar
  23. Broder, A.Z., and A.R. Karlin (1989). Bounds on the cover time, J. Theoretical Probability, 2, 101–120.MathSciNetMATHCrossRefGoogle Scholar
  24. Chen, H-C., and Y. Asau (1974). On generating random variates from an empirical distribution, AILE Trans., No. 2, 6, 163–166.Google Scholar
  25. Cheng, R.C.H. (1977). The generation of Gamma variables with nonintegral shape parameter, Appl. Stat., 26, 71–75.CrossRefGoogle Scholar
  26. Cheng, R.C.H. (1978). Generating beta variates with nonintegral shape parameters, Comm. ACM, 21, 317–322.MathSciNetMATHCrossRefGoogle Scholar
  27. Cheng, R.C.H., and G.M. Feast (1979). Some simple Gamma variate generators,Appl. Statist., 28, 290–295.MATHGoogle Scholar
  28. Cheng, R.C.H., and G.M. Feast (1980). Gamma variate generators with increased shape parameter range, Comm. ACM, 23, 389–394.CrossRefGoogle Scholar
  29. Devroye, L. (1981). The computer generation of Poisson random variables, Computing, 26, 197–207.MathSciNetMATHCrossRefGoogle Scholar
  30. Devroye, L. (1986). Non-Uniform Random Variate Generation, Springer-Verlag, New York.MATHGoogle Scholar
  31. Devroye, L. and A. Naderisamani (1980). A binomial variate generator, Tech. Rep., School of Computer Science, McGill University, Montreal.Google Scholar
  32. Dieter, U. (1982). An alternate proof for the representation of discrete distributions by equiprobable mixtures, J. Appl. Prob., 19, 869–872.MathSciNetMATHCrossRefGoogle Scholar
  33. Fishman, G.S., and L.R. Moore (1984). Sampling from a discrete distribution while preserving monotonicity, Amer. Statist., 38, 219–223.Google Scholar
  34. Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Volume II, second ed., Wiley, New York.MATHGoogle Scholar
  35. Fisher, R.A., and E.A. Cornish (1960). The percentile points of distributions having known cumulants, Technometrics, 2, 209–225.MATHCrossRefGoogle Scholar
  36. Fishman, G.S. (1976). Sampling from the gamma distribution on a computer, Comm. ACM, 19, 407–409.MathSciNetMATHCrossRefGoogle Scholar
  37. Fishman, G.S. (1978). Principles of Discrete Event Simulation, Wiley, New York.MATHGoogle Scholar
  38. Fishman, G.S. (1979). Sampling from the binomial distribution on a computer, J. Amer. Statist. Assoc., 74, 418–423.MathSciNetMATHGoogle Scholar
  39. Fishman, G.S. (1996). Monte Carlo: concepts, Algorithms and Applications, Springer-Verlag, New York.MATHGoogle Scholar
  40. Fishman, G.S., and L.R. Moore (1986). An exhaustive analysis of multiplicative congruential random number generators with modulus 231–1, SIAM J. Sci. Stat. Comput., 7, 24–45.MathSciNetMATHCrossRefGoogle Scholar
  41. Fishman, G.S., and L.S. Yarberry (1993). Generating a sample from a k-cell table with changing probabilities in O(log2 k) time, ACM Trans. Math. Software, 19, 257–261.MATHCrossRefGoogle Scholar
  42. Gerontidis, I., and R.L. Smith (1982). Monte Carlo generation of order statistics from general distributions, Appl. Statist, 31, 238–243.MATHCrossRefGoogle Scholar
  43. Hastings, C., Jr. (1955). Approximations for Digital Computers, Princeton University Press, Princeton, NJ.Google Scholar
  44. Hoeffding, W. (1940). Masstabinvariante Korrelationstheorie, Schriften des Mathematischen Instituts und des Instituts fürAngewandte, Mathematik der UniversitätBerlin, 5, 197–233.Google Scholar
  45. Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58, 13–29.MathSciNetMATHCrossRefGoogle Scholar
  46. Hörmann, W., and G. Derflinger (1993). A portable random number generator well suited for the rejection method, ACM Transactions on Mathematical Software, 19, 489–495.MATHCrossRefGoogle Scholar
  47. Johnson, N.L., S. Kotz, and N. Balakrishnan (1994). Continuous Univariate Distributions, Volume II, second edition, Wiley, New York.MATHGoogle Scholar
  48. Kachitvichyanukul, V., and B. Schmeiser (1985). Computer generation of hypergeometric random variates, J. Statist. Comp. Simul., 22, 127–145.MATHCrossRefGoogle Scholar
  49. Kachitvichyanukul, V., and B. Schmeiser (1988). Binomial random variate generation, Comm. ACM, 31, 216–222.MathSciNetCrossRefGoogle Scholar
  50. Kinderman, A.J., and J.F. Monahan (1977). Computer generation of random variables using the ratio of uniform deviates, ACM Trans. Math. Software, 3, 257–260.MATHCrossRefGoogle Scholar
  51. Kinderman, A.J., and J.F. Monahan (1980). New methods for generating Student’s t and Gamma variables, Computing, 25, 369–377.MathSciNetMATHCrossRefGoogle Scholar
  52. Knuth, D. (1973). The Art of Computer Programming Sorting and Searching, Addison-Wesley, Reading, MA.Google Scholar
  53. Kronmal, Richard A., and A.V. Peterson, Jr. (1979). On the alias method for generating random variables from a discrete distribution, The Amer. Statist., 4, 214–218.MathSciNetGoogle Scholar
  54. Kulkarni, V.G. (1990). Generating random combinatorial objects, J. Algorithms, 11, 185–207.MathSciNetMATHCrossRefGoogle Scholar
  55. Lurie, D., and H.O. Hartley (1972). Machine-generation of order statistics for Monte Carlo computations, The Amer. Statist., 26, 26–27.Google Scholar
  56. MacLaren, M.D., G. Marsaglia, and T.A. Bray (1964). A fast procedure for generating exponential random variables, Comm. ACM, 7, 298–300.MATHCrossRefGoogle Scholar
  57. Marsaglia, G. (1984). The exact-approximation method for generating random variables in a computer, J. Amer. Statist., 79, 218–221.MathSciNetMATHCrossRefGoogle Scholar
  58. Marsaglia, G., K. Ananthanarayanan, and N.J. Paul (1976). Improvements on fast methods for generating normal random variables, Inf. Proc. Letters, 5, 27–30.MathSciNetMATHCrossRefGoogle Scholar
  59. Marsaglia, G., M.D. MacLaren, and T.A. Bray (1964). A fast procedure for generating normal random variables, Comm. ACM, 7, 4–10.MATHCrossRefGoogle Scholar
  60. Marsaglis, G., A. Zaman, and J. Marsaglia (1994) Rapid evaluation of the inverse of the normal distribution function, Statistics and Probability Letters, 19, 259–266.MathSciNetCrossRefGoogle Scholar
  61. Minh, D.L. (1988). Generating gamma variates, ACM Trans. on Math. Software, 14, 261–266.MathSciNetMATHCrossRefGoogle Scholar
  62. Moro, B. (1995). The full Monte, Risk, 8, 57–58.Google Scholar
  63. Ross, S. and Z. Schechner (1986). Simulation uses of the exponential distribution, Stochastic Programming, M. Lucertini, ed., in Lecture Notes in Control and Information Sciences, Springer-Verlag, New York, 76, 41–52.Google Scholar
  64. Schmeiser, B.W., and A.J.G. Babu (1980). Beta variate generation via exponential majorizing functions, Oper. Res., 28, 917–926.MathSciNetMATHCrossRefGoogle Scholar
  65. Schmeiser, B.W., and A.J.G. Babu (1980). Beta variate generation via exponential majorizing functions, Errata: Oper. Res., 31, 1983, 802.MathSciNetGoogle Scholar
  66. Schmeiser, B.W., and R. Lal (1980). Squeeze methods for generating gamma variates, J. Amer. Statist. Assoc. 75, 679–682.MathSciNetMATHCrossRefGoogle Scholar
  67. Schmeiser, B.W., and V. Kachitvichyanukul (1981). Poisson random variate generation, Res. Memo. 8184, School of Industrial Engineering, Purdue University.Google Scholar
  68. Schreider, Y.A. (1964). Method of Statistical Testing, Elsevier, Amsterdam.Google Scholar
  69. Schucany, W.R. (1972). Order statistics in simulation, J. Statist. Comput. and Simul., 1, 281–286.MathSciNetMATHCrossRefGoogle Scholar
  70. Stadlober, E. (1982). Generating Student’s t variates by a modified rejection method, Probability and Statistical Inference, W. Grossmann, G.Ch. Plug, and W. Wertz, eds., Reidel, Dordrecht, Holland, 349–360.Google Scholar
  71. Stadlober, E. (1989). Sampling from Poisson, binomial and hypergeometric distributions: ratio of uniforms as a simple and fast alternative, Math. Statist. Sektion 303, Forschungsgesellschaft Joanneum, Graz, Austria.Google Scholar
  72. Stadlober, E. (1991). Binomial random variate generation: a method based on ratio of uniforms, The Frontiers of Statistical Computation, Simulation, and Modeling, volume 1; P.R. Nelson, E.J. Dudewicz, A. Öztürk, E.C. van der Meulen, editors, American Sciences Press, Syracuse, NY, 93–112.Google Scholar
  73. von Neumann, J. (1951). Various techniques used in connection with random digits, Monte Carlo Method, Applied Mathematics Series 12, National Bureau of Standards, Washington, D.C.Google Scholar
  74. Walker, A.J. (1974). New fast method for generating discrete random numbers with arbitrary frequency distributions, Electronic Letters, 10, 127–128.CrossRefGoogle Scholar
  75. Walker, A.J. (1977). An efficient method for generating discrete random variables with general distributions, ACM Trans. on Math. Software, 3, 253–256.MATHCrossRefGoogle Scholar
  76. Wilks, S.S. (1962). Mathematical Statistics, Wiley, New York.MATHGoogle Scholar
  77. Wong, C.K. and M.C. Easton (1980). An efficient method for weighted sampling without replacement, SIAM J. of Comput., 9, 111–113.MathSciNetMATHCrossRefGoogle Scholar
  78. Zechner, H., and E. Stadlober (1993). Generating Beta variates via patchwork rejection, Computing, 50, 1–18.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • George S. Fishman
    • 1
  1. 1.Department of Operations ResearchUniversity of North Carolina at Chapel HillChapel HillUSA

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