Behavior of Binomial Distribution near Its Median

We study the behavior of the cumulative distribution function of a binomial random variable with parameters n and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b{\text{/}}(n + c)$$\end{document} at the point b – 1 for positive integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b \leqslant n$$\end{document} and real \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c \in [0,\;1]$$\end{document}. Our results can be applied directly to the well-known problem about small deviations of sums of independents random variables from their expectations. Moreover, we answer the question about the monotonicity of the Ramanujan function for the binomial distribution posed by Jogdeo and Samuels in 1968.

Let ξ be a nonnegative integer random variable. The median of ξ is the smallest nonnegative integer such that . For a Poisson random variable with a positive integer parameter b, it is known that [1], which answers the above question. How close is the probability to ?

Ramanujan conjectured [2] that
This conjecture was proved independently by Szegő in [3] and Watson in [4]. Since then the behavior of the function y b has been well studied. In 1913, in his letter to Hardy, Ramanujan made another conjecture: where . This conjecture was proved in 1995 by Flajolet et al. [5]. In 2003, Alm [6] showed that decreases, and, in 2004, Alzer [7] strengthened Ramanujan's conjecture: where moreover, the indicated bounds are sharp.

BINOMIAL DISTRIBUTION AND SAMUELS' PROBLEM
Let be a binomial random variable with parameters n and b/n, where are positive integers. It is well known that converges in distribution to a random variable as . Accordingly, it is natural to expect that the properties of the Poisson distribution described in Section 1 hold for the binomial distribution for sufficiently large n. However, can the same questions be answered for all n?

Moreover, for all for and
Additionally, it was noted in [9] that for all sufficiently large n, but the authors failed to improve this result. Now we consider a binomial random variable with parameters n and , where are positive integers and . Define . The study of the monotonicity of with respect to b is motivated by the well-known problem of small deviation inequality posed by Samuels [10], which can be formulated as follows: find the minimum of over all sets of independent nonnegative random variables with an identical expectation equal to 1. This problem is still unsolved. Nevertheless, it is known that optimal random variables are quantities taking two values with probability 1 (i.e., with two atoms). If the consideration is restricted to identically distributed random variables with two atoms, then the original problem is reduced to analyzing the monotonicity of with respect to b.
The above-mentioned result of Szegö and Watson implies that increases. Since < b), it follows that for sufficiently large n. On the other hand, for example, for and c = 0, we have . Thus, the monotonicity of (regarded as a function of b) changes with increasing n.

NEW RESULTS
We have been able to solve the problem posed by Jogdeo and Samuels concerning the monotonicity of with respect to b.

Theorem 2. Let
. Then there exists such that, for all , the following assertions are true: Additionally, we have examined the function for monotonicity with respect to b. Theorem 3. The following assertions hold:

For all
, it is true that .
3. If and , then . Unfortunately, a complete result was obtained only for c = 0 and c = 1. Nevertheless, we found an asymptotic threshold after which monotonicity changes.
Theorem 4. For all positive δ and , sufficiently large n, and integer , the following assertions hold: The analysis of these expressions is reduced to examining the behavior of the function g(z) = on . Its behavior can be studied using the Taylor formula (up to the fifth term) with a Lagrange remainder and the following convenient representation of the derivatives of g: FUNDING This work was supported by the Russian Science Foundation, project no. 21-71-10092.