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 $n$.
Best Answer
To elaborate on @user1551 comment, under independence, you could use the law of iterated expectations to derive what you need. First, the conditional expectation of $Z$ on fixing $Y=y$ is \begin{eqnarray*} E \left[ Z|Y = y \right] & = & E \left[ \sum_{i = 1}^y X_i \right]\\ & = & yE \left[ X_i \right] \end{eqnarray*} This allows to compute $E[Z]$ \begin{eqnarray*} E \left[ Z \right] & = & E_Y \left[ E \left[ Z|Y = y \right] \right]\\ & = & E_Y \left[ yE \left[ X_i \right] \right]\\ & = & E \left[ Y \right] E \left[ X_i \right] \end{eqnarray*}
For the variance of $Z$, you could exploit formulas for conditional variances (such as here).
You could apply a similar argument in order to derive the cumulative distribution of $Z$. \begin{eqnarray*} \Pr \left[ Z \leqslant z \right] & = & E_Y \left[ \Pr \left[ Z \leqslant z|y \right] \right] \end{eqnarray*}