To get the 2 first moments of a composed poisson variable I get that E(e^tw) where w is ΣXi is Σ[E(e^tXi)*e^-λt *(λt)^n/n!]. So I transform E(e^tXi) to фx(t)
Substituting
Σ[(фx(t))^n *e^-λt *(λt)^n/n!]
and then it transforms to
exp{λt(фx(t)-1)}
How did I get to this last step? There is any tranformation or definition that i should know to do it? If someone can explain me step by step how should I do it to get to that last ecuation please