The distribution of the de Moivre experiment

In this paper, we focus on de Moivre random experience 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 and the generating functions. Then we establish that for $s\in \mathbb{N}$, the bi${}^s$nomial distribution converges to a limiting Poisson and normal distributions when $n\to \mathrm{\infty }\mathrm{.}$