The distribution of the de Moivre experiment

Abstract

In this paper, we focus on the de Moivre random experiment, which allows us to introduce the $s$-Bernoulli distribution and the bi$^{s}$nomial distribution. We present some probabilistic properties such as the expectation, the variance, the skewness and kurtosis coefficients, the moments, the cumulants and the generating functions; the moments are expressed through extended Eulerian numbers and partial Bell polynomials, and the cumulants through logarithmic polynomials. Then we establish that, for an integer $s\geq 2$, the bi$^{s}$nomial distribution converges to a Poisson limiting distribution when the expectations converge, and that, for fixed parameters with $p\neq q$, it satisfies a central limit theorem as $n\rightarrow\infty$.