Bounds for the error of the Gaussian approximation for the binomial distribution are stated, depending from the probability of success and the number n of observations. As a consequence, the upper bound for the absolute constant in the Berry-Esseen inequality for identically distributed random variables, taking two values, is deduced which differs from asymptotical one slightly more than 0.01. The following idea is realized in the work. We can obtain sharp bounds for sufficiently large n. The main purpose of the paper is to prove just these bounds. As to bounded number of observations, computations with the help of the computer must be produced. This part of investigations is developed by our pupils K.V. Mikhailov and A.S. Kondric.

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It is shown that the absolute constant in the Berry-Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if 1 ≤ n ≤ 500000, where n is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009-2011, an upper bound is obtained for the absolute constant in the Berry-Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.06%. As an auxiliary result, we prove a bound in the local Moivre-Laplace theorem which has a simple and explicit form.Despite the best possible result, obtained by J. Schulz in 2016, we propose our approach to the problem of finding the absolute constant in the Berry-Esseen inequality for two-point distributions since this approach, combining analytical methods and the use of computers, could be useful in solving other mathematical problems.Keywords Optimal value of absolute constant in Berry-Esseen inequality, binomial distribution, numerical methods 2010 MSC 60F05, 65-04

In this chapter, we propose a probabilistic model for train delay propagation. There are deduced formulas for the probability distributions of arrival headways and knock-on delays depending on distributions of the primary delay duration and the departure headways. We prove some key mathematical statements. The obtained formulas allow to predict the frequency of train arrival delays and to determine the optimal traffic adjustments. Several important special cases of initial probability distributions are considered. Results of the theoretical analysis are verified by comparison with statistical data on the train traffic at the Russian railways.