Advertisement

Error Bounds for Trapezoid Type Quadrature Rules with Applications for the Mean and Variance

  • Pietro Cerone
  • Sever S. Dragomir
  • Eder KikiantyEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 91)

Abstract

In this paper, we establish some inequalities of trapezoid type to give tight bounds for the expectation and variance of a probability density function. The approach is also demonstrated for higher order moments.

Keywords

Probability Density Function Error Bound Bounded Variation Type Inequality Gaussian Random Variable 
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.

References

  1. 1.
    Barnett, N.S., Cerone P., and Dragomir, S.S.: Inequalities for distributions on a finite interval. Advances in Mathematical Inequalities Series. Nova Science Publishers Inc., New York (2008)zbMATHGoogle Scholar
  2. 2.
    Brnetić, I. and Pečarić, J.: On an Ostrowski type inequality for a random variable. Math. Inequal. Appl. 3, No. 1, 143–145 (2000)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cacoullos, T. Variance inequalities, characterizations, and a simple proof of the central limit theorem. Data analysis and statistical inference, 27–32 (1992)Google Scholar
  4. 4.
    Cacoullos, T. and Papadatos, N. and Papathanasiou, V.: Variance inequalities for covariance kernels and applications to central limit theorems. Teor. Veroyatnost. i Primenen. 42, No. 1, 195–201 (1997)Google Scholar
  5. 5.
    Cerone P., and Dragomir S.S.: Trapezoid type rules from an inequalities point of view. Handbook of Analytic Computational Methods in Applied Mathematics. Anastassiou G. (Ed.), pp. 65–134. CRC Press, N.Y. (2000)Google Scholar
  6. 6.
    Chang, W-Y. and Richards, D.St.P.: Variance inequalities for functions of multivariate random variables. Contemp. Math. 234, 43–67 (1999)Google Scholar
  7. 7.
    Chernoff, H.: A note on an inequality involving the normal distribution. Ann. Probab. 9, No. 3, 533–535 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dharmadhikari, S. W. and Joag-Dev, K.: Upper bounds for the variances of certain random variables. Comm. Statist. Theory Methods 18, No. 9, 3235–3247 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Houdré, C. and Kagan, A.: Variance inequalities for functions of Gaussian variables. J. Theoret. Probab. 8, No. 1, 23–30 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Prakasa Rao, B. L. S. and Sreehari, M.: Chernoff-type inequality and variance bounds. J. Statist. Plann. Inference 63, No. 2, 325–335 (1997)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Pietro Cerone
    • 1
  • Sever S. Dragomir
    • 2
    • 3
  • Eder Kikianty
    • 4
    Email author
  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityBundooraAustralia
  2. 2.School of Engineering and ScienceVictoria UniversityMelbourneAustralia
  3. 3.School of Computational and Applied MathematicsUniversity of the WitwatersrandWitwatersrandSouth Africa
  4. 4.Department of MathematicsUniversity of JohannesburgAuckland ParkSouth Africa

Personalised recommendations