1. [4] Let $Z_1, Z_2, \dots$ be independent, standard normally distributed random variables.
For a given $n \in \mathbb{N}$, let $N$ be the discrete random variable such that
$\qquad P(N = 1) = \frac{1}{n}, \qquad P(N = n) = 1 - \frac{1}{n}$.
Find the moment generating function of
$\qquad Y = \sum_{i=1}^N \frac{Z_i + 1/N}{\sqrt{N}}.$
To which known distribution does $Y$ converge for $n \to \infty$?