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arXiv · 1807.06527

On monotonicity of Ramanujan function for binomial random variables

Daniil Dmitriev·Maksim Zhukovskii
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Abstract

For a binomial random variable $\xi$ 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 ${\sf P}(\xi=b)$ and $1/2-{\sf P}(\xi<b)$. 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.

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Daniil Dmitriev, Maksim Zhukovskii. 2018-07-17. On monotonicity of Ramanujan function for binomial random variables. https://arxiv.org/abs/1807.06527

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