Probability and Statistics

  • Peter E. Kloeden
  • Eckhard Platen
Chapter
Part of the Applications of Mathematics book series (SMAP, volume 23)

Abstract

The basic concepts and results of probability and stochastic processes needed later in the book are reviewed here. The emphasis is descriptive and PC-Exercises (PC= Personal Computer), based on pseudo-random number generators introduced in Section 3, are used extensively to help the reader to develop an intuitive understanding of the material. Statistical tests are discussed briefly in the final section.

Keywords

Covariance Dinates Clarification Suffix 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical Notes

  1. Parzen, E. (1962). Stochastic Processes. Holden-Day, San Francisco.MATHGoogle Scholar
  2. Papoulis, A. (1984). Probability, Random Variables and Stochastic Processes ( 2nd ed. ). McGraw-Hill Series in Electrical Engineering. Communications and Information Theory. McGraw-Hill, New York.MATHGoogle Scholar
  3. Ash, R. B. (1970). Basic Probability Theory. Wiley, New York.MATHGoogle Scholar
  4. Karlin, S. and H. M. Taylor (1975). A First Course in Stochastic Processes ( 2nd ed. ). Academic Press, New York.MATHGoogle Scholar
  5. Karlin, S. and H. M. Taylor (1975). A First Course in Stochastic Processes ( 2nd ed. ). Academic Press, New York.MATHGoogle Scholar
  6. Karlin, S. and H. M. Taylor (1981). A Second Course in Stochastic Processes. Academic Press, New York.MATHGoogle Scholar
  7. Chung, K. L. (1975). Elementary Probability with Stochastic Processes. Springer.Google Scholar
  8. Bywater, R. J. and P. M. Chung (1973). Turbulent flow fields with two dynamically significant scales. AIAA papers, 73–646, 991–10.Google Scholar
  9. Shiryayev, A. N. (1984). Probability. Springer.Google Scholar
  10. Ripley, B. D. (1983b). Stochastic Simulation. Wiley, New York.Google Scholar
  11. Ripley, B. D. (1983a). Computer generation of random variables: A tutorial letter. Internat. Statist. Rev. 45, 301–319.MathSciNetCrossRefGoogle Scholar
  12. Ermakov, S. M. (1975). Die Monte-Carlo-Methode und verwandte Fragen. Hochschulbücher für Mathematik, Band 72. VEB Deutscher Verlag der Wissenschaften, Berlin. (in German) Translated from Russian by E. Schincke and M. Schleiff.Google Scholar
  13. Yakowitz, S. J. (1977). Computational Probability and Simulation. Addison Wesley, Reading, MA.Google Scholar
  14. Rubinstein, R. Y. (1981). Simulation and the Monte Carlo Method. Wiley Ser. Probab. Math. Statist. Wiley, New York.Google Scholar
  15. Morgan, B. J. (1984). Elements of Simulation. Chapman and Hall, London.Google Scholar
  16. Ross, S. M. (1990). A Course in Simulation. Macmillan.Google Scholar
  17. Fishman, G. S. (1996). Monte Carlo: Concepts, Algprithms and Applications. Springer Ser. Oper. Res. Springer.Google Scholar
  18. Ermakov, S. M. and Mikhailov (1982). Statistical Modeling ( 2nd ed. ). Nauka, Moscow.Google Scholar
  19. J. E. Gentle Random Number Generation and Monte Carlo Methods. Springer Ser. Statist Comput SpringerGoogle Scholar
  20. Box, G. and M. Muller (1958). A note on the generation of random normal variables. Ann. Math. Statist. 29, 610–611.MATHCrossRefGoogle Scholar
  21. Box, G. and M. Muller (1958). A note on the generation of random normal variables. Ann. Math. Statist. 29, 610–611.MATHCrossRefGoogle Scholar
  22. Marsaglia, G. and T. A. Bray (1964). A convenient method for generating normal variables. SIAM Rev. 6, 260–264.MathSciNetMATHCrossRefGoogle Scholar
  23. Brent, R. P. (1974). A Gaussian pseudo number generator. Commun. Assoc. Comput. Mach. 17, 704–706.MATHGoogle Scholar
  24. Petersen, W. P. (1988). Some vectorized random number generators for uniform, normal and Poisson distributions for CRAY X-MP. J. Supercomputing 1, 318335.Google Scholar
  25. Clements, D. J. and B. D. O. Anderson (1973). Well behaved Ito equations with simulations that always misbehave. IEEE Trans. Automat. Control 18, 676–677.MATHCrossRefGoogle Scholar
  26. Eichenauer, J. and J. Lehn (1986). A nonlinear congruential pseudo random number generator. Statist. Papers 27(4), 315–326.Google Scholar
  27. Niederreiter, H. (1988). Remarks on nonlinear pseudo random numbers. Metrika 35, 321–328.MathSciNetMATHCrossRefGoogle Scholar
  28. Eichenauer-Herrmann, J. (1991). Inverse congruential pseudo random number generators avoid the planes. Math. Comp. 56, 297–301.MathSciNetMATHCrossRefGoogle Scholar
  29. Cinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, N. J.Google Scholar
  30. Shiryayev, A. N. (1984). Probability. Springer.Google Scholar
  31. Parzen, E. (1962). Stochastic Processes. Holden-Day, San Francisco.MATHGoogle Scholar
  32. Jazwinski, A. (1970). Stochastic Processes and Filtering Theory. Academic Press, London.Google Scholar
  33. van Kampen, N. G. (1981a). Ito versus Stratonovich. J. Statist. Phys. 24(1), 175187.Google Scholar
  34. van Kampen, N. G. (1981b). Stochastic Processes in Physics and Chemistry,Volume 888 of Lecture Notes in Math. North Holland.Google Scholar
  35. Skorokhod, A. V. and N. P. Slobodenjuk (1970). Limit Theorems for Random Walks. Naukova Dumka, Kiev. (in Russian).Google Scholar
  36. Skorokhod, A. V. (1982). Studies in the Theory of Stochastic Processes. Dover, New York.Google Scholar
  37. Stroock, D. W. (1982). Lectures on Topics in Stochastic Differential Equations,Volume 68 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Bombay. Springer.Google Scholar
  38. Stroock, D. W. and S. R. S. Varadhan (1982). Multidimensional Diffusion Processes, Volume 233 of Grundlehren Math. Wiss. Springer.Google Scholar
  39. Papoulis, A. (1984). Probability, Random Variables and Stochastic Processes ( 2nd ed. ). McGraw-Hill Series in Electrical Engineering. Communications and Information Theory. McGraw-Hill, New York.MATHGoogle Scholar
  40. Wong, E. W. (1971). Stochastic Processes in Information and Dynamical Systems. McGraw-Hill, New York.MATHGoogle Scholar
  41. Wong, E. W. and B. Hajek (1985). Stochastic Processes in Engineering Systems. Springer.Google Scholar
  42. Wong, E. W. and M. Zakai (1965a). On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Statist. 36, 1560–1564.MathSciNetMATHCrossRefGoogle Scholar
  43. Kleijnen, J. P. C. (1975). Statistical Techniques in Simulation. Part II, Volume 9 of Statistics: Textbooks and Monographs. Marcel Dekker, New York.Google Scholar
  44. Kleijnen, J. P. C. (1974). Statistical Techniques in Simulation. Part I, Volume 9 of Statistics: Textbooks and Monographs. Marcel Dekker, New York.Google Scholar
  45. Dashevski, M. I. and R. S. Liptser (1966). Simulation of stochastic differential equations connected with the disorder problem by means of analog computer. Automat. Remote Control 27, 665–673. (in Russian).Google Scholar
  46. Jacod, J. and A. N. Shiryaev (1987). Limit Theorems for Stochastic Processes. Springer.Google Scholar
  47. Liptser, R. and A. Shiryaev (1977). Statistics of Random Processes: I. General Theory, Volume 5 of Appl. Math. Springer.Google Scholar
  48. Liptser, R. and A. Shiryaev (1977). Statistics of Random Processes: I. General Theory, Volume 5 of Appl. Math. Springer.Google Scholar
  49. Liptser, R. and A. Shiryaev (1978). Statistics of Random Processes: II. Applications, Volume 6 of Appl. Math. Springer.Google Scholar
  50. Groeneveld, R. A. (1979). An Introduction to Probability and Statistics using BASIC. Marcel-Dekker, New York.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Peter E. Kloeden
    • 1
  • Eckhard Platen
    • 2
  1. 1.Fachbereich MathematikJohann Wolfgang Goethe-UniversitätFrankfurt am MainGermany
  2. 2.School of Mathematical Sciences and School of Finance & EconomicsUniversity of Technology, SydneyBroadwayAustralia

Personalised recommendations