A Note on Weighted Count Distributions

As particular case of the Radon-Nikodym theorem we show that any count distribution is a weighted Poisson distribution (WPD) and, more generally, a weighted version of any other count distribution with suitable support. Several properties and consequences of this result are then given. With a condition on their supports, we deduce that two count distributions are connected by the weightening operator which commutes with some classical transfor-mations. The notion of dual distribution, initially introduced for WPD, is also extended to any count distribution and a practical interpretation of its meaning is given. Finally, some concluding remarks and discussions are made for practical purposes