The median of gamma distribution and a related Ramanujan sequence

The Ramanujan Journal - Tập 44 - Trang 75-88 - 2016
Chao-Ping Chen1
1School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, China

Tóm tắt

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 $$e^n$$ .

Tài liệu tham khảo

Adell, J.A., Jodrá, P.: Sharp estimates for the median of the \(\Gamma (n+1, 1)\) distribution. Stat. Probab. Lett. 71, 185–191 (2005) 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) Adell, J.A., Jodrá, P.: On the complete monotonicity of a Ramanujan sequence connected with \(e^n\). Ramanujan J. 16, 1–5 (2008) Adell, J.A., Alzer, H.: Inequalities for the median of the gamma distribution. J. Comput. Appl. Math. 232, 481–495 (2009) 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) Alzer, H.: On Ramanujan’s inequalities for exp(k). J. Lond. Math. Soc. 69, 639–656 (2004) Alzer, H.: Proof of the Chen–Rubin conjecture. Proc. R. Soc. Edinb. 135A, 677–688 (2005) Alzer, H.: A convexity property of the median of the gamma distribution. Stat. Probab. Lett. 76, 1510–1513 (2006) Berg, C., Pedersen, H.L.: The Chen-Rubin conjecture in a continuous setting. Methods Appl. Anal. 13, 63–88 (2006) Berg, C., Pedersen, H.L.: Convexity of the median in the gamma distribution. Ark. Mat. 46, 1–6 (2008) Bracken, P.: A function related to the central limit theorem. Comment. Math. Univ. Carol. 39, 765–775 (1998) Chen, J., Rubin, H.: Bounds for the difference between median and mean of gamma and Poisson distributions. Stat. Probab. Lett. 4, 281–283 (1986) 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) Choi, K.P.: On the medians of gamma distributions and an equation of Ramanujan. Proc. Am. Math. Soc. 121, 245–251 (1994) Flajolet, P., Grabner, P.J., Kirschenhofer, P., Prodinger, H.: On Ramanujan’s Q-function. J. Comput. Appl. Math. 58, 103–116 (1995) Karamata, J.: Sur quelques problèmes posés par Ramanujan. J. Indian Math. Soc. 24, 343–365 (1960) Koumandos, S.: A Bernstein function related to Ramanujan’s approximations of \(\exp (n)\). Ramanujan J. 30, 447–459 (2013) Marsaglia, J.C.W.: The incomplete Gamma function and Ramanujan’s rational approximation to \(e^x\). J. Stat. Comput. Simul. 24, 163–169 (1986) Mortici, C.: New approximations of the gamma function in terms of the digamma function. Appl. Math. Lett. 23, 97–100 (2010) Mortici, C.: Product approximations via asymptotic integration. Am. Math. Monthly 117, 434–441 (2010) Ramanujan, S.: Question 294. J. Indian Math. Soc. 3, 128 (1911) Ramanujan, S.: On Question 294. J. Indian Math. Soc. 4, 151–152 (1912) Ramanujan, S.: Collected Papers. Chelsea, New York (1962) Szegö, G.: Über einige von S. Ramanujan gestelle Aufgaben. J. Lond. Math. Soc. 3, 225–232 (1928) Volkmer, H.: Factorial series connected with the Lambert function, and a problem posed by Ramanujan. Ramanujan J. 16, 235–245 (2008) Watson, G.N.: Theorems stated by Ramanujan (V): approximations connected with \(e^x\). Proc. Lond. Math. Soc. 29, 293–308 (1929)