J. Appl. Prob. 48, 877–884 (2011) Printed in England  Applied Probability Trust 2011 ON THE CONVOLUTION OF HETEROGENEOUS BERNOULLI RANDOM VARIABLES MAOCHAO XU, ∗ Illinois State University N. BALAKRISHNAN, ∗∗ McMaster University Abstract In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002). Keywords: Bernoulli; heterogeneous variable; likelihood ratio order; majorization; reversed hazard rate order 2010 Mathematics Subject Classification: Primary 60E15; 62N05; 62G30; 62D05 1. Introduction The Bernoulli distribution is one of the most fundamental distributions in statistics, and has found key applications in engineering, actuarial science, operations research, and reliability theory. Recently, much attention has been paid to the study of convolutions of independent random variables, such as exponential, gamma, Weibull, and geometric. We refer the reader to Zhao and Balakrishnan (2010), Mao et al. (2010), Kochar and Xu (2011), and the references therein for some related developments. Let X p 1 ,...,X p n be a sequence of independent Bernoulli random variables with parameters p 1 ,...,p n , respectively. The convolution of independent Bernoulli random variables was first considered in Hoeffding (1956), who showed that P  n  i =1 X p i ≤ k  ≤ k  j =0  n j  ¯ p j (1 -¯ p) n-j for 0 ≤ k ≤ n ¯ p - 1 and P  n  i =1 X p i ≤ k  ≥ k  j =0  n j  ¯ p j (1 -¯ p) n-j for n ¯ p ≤ k ≤ n, where ¯ p = ∑ n i =1 p i /n. As a consequence, Hoeffding further showed that, for any two integers b and c such that 0 ≤ b ≤ n ¯ p ≤ c ≤ n, P  b ≤ n  i =1 X p i ≤ c  ≥ c  j =b  n j  ¯ p j (1 -¯ p) n-j . Received 14 September 2010; revision received 24 February 2011. ∗ Postal address: Department of Mathematics, Illinois State University, Normal, IL, USA. Email address: mxu2@ilstu.edu ∗∗ Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada. 877 at https://www.cambridge.org/core/terms. https://doi.org/10.1239/jap/1316796922 Downloaded from https://www.cambridge.org/core. IP address: 91.132.250.239, on 24 Apr 2020 at 08:29:57, subject to the Cambridge Core terms of use, available