Skip to main content
Log in

The median of gamma distribution and a related Ramanujan sequence

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

We consider the continued fraction approximation of the median of gamma distribution.We also consider asymptotic series and continued fraction approximation of a Ramanujan sequence connected with \(e^n\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Adell, J.A., Jodrá, P.: Sharp estimates for the median of the \(\Gamma (n+1, 1)\) distribution. Stat. Probab. Lett. 71, 185–191 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adell, J.A., Jodrá, P.: On a Ramanujan equation connected with the median of the gamma distribution. Trans. Am. Math. Soc. 360, 3631–3644 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adell, J.A., Jodrá, P.: On the complete monotonicity of a Ramanujan sequence connected with \(e^n\). Ramanujan J. 16, 1–5 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Adell, J.A., Alzer, H.: Inequalities for the median of the gamma distribution. J. Comput. Appl. Math. 232, 481–495 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alm, S.E.: Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence. Bernoulli 9, 351–371 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alzer, H.: On Ramanujan’s inequalities for exp(k). J. Lond. Math. Soc. 69, 639–656 (2004)

    Article  MATH  Google Scholar 

  7. Alzer, H.: Proof of the Chen–Rubin conjecture. Proc. R. Soc. Edinb. 135A, 677–688 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Alzer, H.: A convexity property of the median of the gamma distribution. Stat. Probab. Lett. 76, 1510–1513 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Berg, C., Pedersen, H.L.: The Chen-Rubin conjecture in a continuous setting. Methods Appl. Anal. 13, 63–88 (2006)

    MathSciNet  MATH  Google Scholar 

  10. Berg, C., Pedersen, H.L.: Convexity of the median in the gamma distribution. Ark. Mat. 46, 1–6 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bracken, P.: A function related to the central limit theorem. Comment. Math. Univ. Carol. 39, 765–775 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Chen, J., Rubin, H.: Bounds for the difference between median and mean of gamma and Poisson distributions. Stat. Probab. Lett. 4, 281–283 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheng, T.T.: The normal approximation to the Poisson distribution and a proof of a conjecture of Ramanujan. Bull. Am. Math. Soc. 55, 396–401 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  14. Choi, K.P.: On the medians of gamma distributions and an equation of Ramanujan. Proc. Am. Math. Soc. 121, 245–251 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  15. Flajolet, P., Grabner, P.J., Kirschenhofer, P., Prodinger, H.: On Ramanujan’s Q-function. J. Comput. Appl. Math. 58, 103–116 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Karamata, J.: Sur quelques problèmes posés par Ramanujan. J. Indian Math. Soc. 24, 343–365 (1960)

    MathSciNet  MATH  Google Scholar 

  17. Koumandos, S.: A Bernstein function related to Ramanujan’s approximations of \(\exp (n)\). Ramanujan J. 30, 447–459 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Marsaglia, J.C.W.: The incomplete Gamma function and Ramanujan’s rational approximation to \(e^x\). J. Stat. Comput. Simul. 24, 163–169 (1986)

    Article  Google Scholar 

  19. Mortici, C.: New approximations of the gamma function in terms of the digamma function. Appl. Math. Lett. 23, 97–100 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mortici, C.: Product approximations via asymptotic integration. Am. Math. Monthly 117, 434–441 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ramanujan, S.: Question 294. J. Indian Math. Soc. 3, 128 (1911)

    Google Scholar 

  22. Ramanujan, S.: On Question 294. J. Indian Math. Soc. 4, 151–152 (1912)

    Google Scholar 

  23. Ramanujan, S.: Collected Papers. Chelsea, New York (1962)

    Google Scholar 

  24. Szegö, G.: Über einige von S. Ramanujan gestelle Aufgaben. J. Lond. Math. Soc. 3, 225–232 (1928)

    Article  MATH  Google Scholar 

  25. Volkmer, H.: Factorial series connected with the Lambert function, and a problem posed by Ramanujan. Ramanujan J. 16, 235–245 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Watson, G.N.: Theorems stated by Ramanujan (V): approximations connected with \(e^x\). Proc. Lond. Math. Soc. 29, 293–308 (1929)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chao-Ping Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, CP. The median of gamma distribution and a related Ramanujan sequence. Ramanujan J 44, 75–88 (2017). https://doi.org/10.1007/s11139-016-9796-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-016-9796-y

Keywords

Mathematics Subject Classification

Navigation