Chapter 6. Integration

  1. André I. Khuri

Published Online: 26 MAR 2003

DOI: 10.1002/0471394882.ch6

Advanced Calculus with Applications in Statistics, Second Edition

Advanced Calculus with Applications in Statistics, Second Edition

How to Cite

Khuri, A. I. (2002) Integration, in Advanced Calculus with Applications in Statistics, Second Edition, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/0471394882.ch6

Author Information

  1. University of Florida, Gainesville, Florida, USA

Publication History

  1. Published Online: 26 MAR 2003
  2. Published Print: 1 NOV 2002

Book Series:

  1. Wiley Series in Probability and Statistics

Book Series Editors:

  1. David J. Balding,
  2. Peter Bloomfield,
  3. Noel A. C. Cressie,
  4. Nicholas I. Fisher,
  5. Iain M. Johnstone,
  6. J. B. Kadane,
  7. Louise M. Ryan,
  8. David W. Scott,
  9. Adrian F. M. Smith and
  10. Jozef L. Teugels

ISBN Information

Print ISBN: 9780471391043

Online ISBN: 9780471394884



  • Cauchy–Schwarz inequality;
  • Chebyshev's inequality;
  • first negative moment;
  • function of bounded variation;
  • fundamental theorem of calculus;
  • Holder's inequality;
  • improper Riemann integral;
  • Jensen's inequality;
  • Maclaurin's integral test;
  • Minkowski's inequality;
  • Riemann integral;
  • Riemann–Stieltjes integral


This chapter discusses Riemann integration of real-valued functions, including a study of improper Riemann integrals, and convergence of a sequence of Riemann integrals. It also covers the Riemann–Stieltjes integral. Applications in statistics are given in the last section, which includes a discussion on the existence of the first negative moment of a continuous distribution, transformations of continuous random variables, and the Riemann–Stieltjes integral representation of the expected value of a random variable.