# Chapter 6. Integration

Published Online: 26 MAR 2003

DOI: 10.1002/0471394882.ch6

Copyright © 2003 John Wiley & Sons, Inc.

Book Title

## Advanced Calculus with Applications in Statistics, Second Edition

Additional Information

#### 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

#### Publication History

- Published Online: 26 MAR 2003
- Published Print: 1 NOV 2002

#### Book Series:

#### Book Series Editors:

- David J. Balding,
- Peter Bloomfield,
- Noel A. C. Cressie,
- Nicholas I. Fisher,
- Iain M. Johnstone,
- J. B. Kadane,
- Louise M. Ryan,
- David W. Scott,
- Adrian F. M. Smith and
- Jozef L. Teugels

#### ISBN Information

Print ISBN: 9780471391043

Online ISBN: 9780471394884

- Summary
- Chapter

### Keywords:

- 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

### Summary

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.