Poissoniana

$$G(t\,\vert\,{\cal P}_{n,p}) = \sum_{x=0}^{\infty}\frac{e^{t\, x}\, e^{-\lambda}\, \lambda^x}{x!} = e^{-\lambda}\, \sum_{x=0}^{\infty}\frac{\left(\lambda\, e^t\right)^x}{x!}$$

$$= e^{-\lambda}e^{\lambda e^t}\,,$$

da cui

$$(X) = \left.\lambda\, e^{-\lambda}\, e^{\lambda\, e^t}\, e^t\right\vert _{t=0} = \lambda$$

$$(X^2) = \left.\lambda\, e^{-\lambda}\, \left(\lambda\, e^{\lambda\, e^t}\, e^t +e^{\lambda\, e^t}\right)\right\vert _{t=0} = \lambda\, (\lambda+1)$$

$$(X) = \lambda\,.$$