2018/07/17 by Daniil Dmitriev, Dmitriev, Daniil, Maksim Zhukovskii +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1807.06527
arxiv created 2021/04/08 · arxiv updated 2021/04/09
For a binomial random variable ξ with parameters n and b/n, it is well known that the median equals b when b is an integer. In 1968, Jogdeo and Samuels studied the behaviour of the relative difference between \sf P(ξ=b) and 1/2-\sf P(ξ<b). They proved its monotonicity in n and posed a question about its monotonicity in b. This question is motivated by the solved problem proposed by Ramanujan in 1911 on the monotonicity of the same quantity but for a Poisson random variable with an integer parameter b. In the paper, we answer this question and introduce a simple way to analyse the monotonicity of similar functions.