On the nature of the binomial distribution
A sequence of Bernoulli variables \(\{X_i\}\) is considered. The sum of \(n\) those independent variables (a) has the binomial distribution \(B(n,p)\) with parameters \(n\) and \(p\), if \(P(X_i=1)= p\) for every \(i\) (b). This paper demonstates that \(B(n,p)\) arises also when both conditions (a and b) are violated. Especially, the structure of distributions of \(B(2,p)\) and \(B(3,p)\) for \(B(n,p)\) (\(n\) arbitrary) are obtained. The Poisson distribution for a random sum is characterized. This brings out the relationship between the Poisson process and the binomial distributions.
scientific article; zbMATH DE number 1625605