Originally Posted by

**mr fantastic** Well, I guess this is about the last one in the list!

$\displaystyle m_X(t) = \sum_{n=0}^{\infty} e^{nt} \frac{\lambda^n e^{-\lambda}}{n!} = e^{-\lambda} \sum_{n=0}^{\infty} e^{nt} \frac{\lambda^n}{n!}$

$\displaystyle = e^{-\lambda} \sum_{n=0}^{\infty} \frac{(\lambda e^t)^n}{n!}$

using the standard series $\displaystyle \sum_{n=0}^{\infty} \frac{(y)^n}{n!} = e^y$ and substituting $\displaystyle y = \lambda e^t$ *

$\displaystyle = e^{-\lambda} \, e^{\lambda e^t} = e^{\lambda(e^t - 1)}$.

* If you don't like doing this there is another clever way of doing it.