On monotonicity of Ramanujan function for binomial random variables

TL;DR

This paper investigates the monotonicity properties of a specific probability-related function for binomial variables, answering a question posed by Jogdeo and Samuels, and relates it to Ramanujan's earlier work on Poisson variables.

Contribution

The paper answers the open question about the monotonicity of the function in parameter b for binomial variables and introduces a new method for analyzing such monotonicity.

Findings

Confirmed monotonicity in parameter n

Established monotonicity in parameter b

Provided a simple analytical approach for similar functions

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.