The main aim of this article is to characterize and investigate the three parameter exponentiated exponential Poisson probability distribution ${\rm EEP}(α, β, λ)$ by giving explicit closed form expressions for its characteristic function $φ_ξ(t)$ and moment generating function $M_ξ(t)$, and finally, to show that the existing series and integral form expressions for positive integer order moments $\mathbb Eξ^ν, ν\in \mathbb N$ are in fact valid for all $ν>1-α, α>0$.