On sum of 0-1 random variables. I. Univariate case
This article deals with the exact distribution of \(S_ n=X_ 1+...+X_ n\) where \(X_ 1,...,X_ n\) are 0-1 random variables. Independence or exchangeability is not assumed. The distribution of \(S_ n\) is studied by evaluating the factorial moments and then using the fact that the factorial moments determine the distribution. A binomial approximation of the distribution of \(S_ n\) is obtained through orthogonal expansions in terms of orthogonal polynomials in the category of Krawtchouk polynomials. Poisson and normal type approximations are also obtained by exploiting the factorial moments.
scientific article; zbMATH DE number 4005227