I feel something is wrong, but can't place it:
Assume $X_i$ are i.i.d. Poisson distribution with parameter $\lambda$ and define
$$Y = \sum_{i=1}^n X_i $$
$$M_X(t) = \exp((e^t-1)\cdot\lambda)$$
$$M_Y(t) = \Big({\exp((e^t-1)\cdot\lambda)\Big)}^n = \exp((e^t-1)\cdot n\cdot\lambda)$$
where $M_X(\cdot)$ is moment generating function.
so I think $$f_Y(y)=\text{The pdf of }Y = {e^{-n\lambda}(n\lambda)^y\over y!}$$
but in my textbook $$f_Y(y) = {e^{-n\lambda}(\lambda)^y\over y!}$$
I don't understand it!
Is there something wrong in my solution?
or
is the textbook wrong?