Prob­a­bil­ity Gen­er­at­ing Func­tion for the Pois­son Distribution

Let p(y) be the prob­a­bil­ity gen­er­at­ing func­tion for Y.

\begin{aligned} p_y(t) = E(t^y) &= {\sum_{y=0}^{\infty}} t^y p(y) \\ &= \sum_{y=0}^{\infty} t^{y} \lambda^{y}e^{-\lambda} \\ &= e^{-\lambda}\sum_{y=0}^{\infty} {{t^{y}\lambda^{y}} \over {y!}} \\ &= e^{-\lambda}\sum_{y=0}^{\infty} {(t\lambda)^{y} \over {y!}} ~~~~\mbox{Power Series}\\&= e^{-\lambda}e^{t\lambda} \\&= e^{t\lambda - \lambda} \\&= e^{\lambda(t-1)}\end{aligned}

Expected Value of the Pois­son Distribution

\begin{aligned}E(Y) = \mu_{[1]} &= {\frac{d(P_{y}(t)}{dt}}\left.{\!\!\frac{}{}}\right|_{t=1}\\&= {\frac{d(e^{\lambda(t-1)})}{dt}}\left.{\!\!\frac{}{}}\right|_{t=1}\\&= {e^{\lambda(t-1)} \cdot \lambda}\left.{\!\!\frac{}{}}\right|_{t=1}\\&= \lambda\end{aligned}