Consider a sequence of independent and identically distributed random variables $X_1, X_2, \ldots$ such that $$ \mathbb P(X_i\geqslant k)=\prod_{\ell=1}^{k-2}\frac{n-\ell}n \ \textrm{ for every } 2\leqslant k\leqslant n+1. $$
Consider also $$Z= \sum_{i=1}^Y X_i ,$$
where $Y$ follows a geometric distribution with success probability $1/n$.
What is the mean and variance of $Z$ and is it possible to calculate its full distribution? I am particularly interested in what happens for large $n$.