I am stuck in a problem, and I can't think of a next step to find the solution. The question is the following:
Suppose an urn has $k$ balls, numbered from $1$ to $k$, $k \in \mathbb{N}$. A sample of $m$ balls is withdrawn, $m \in \mathbb{N}^ \ : \ 1\leq m < k$ and the numbers of such sample are written down. Then the $m$ balls are returned to the urn and another sample is withdrawn, with $m$ balls again and registering the numbers. Let $X_n$ be the number of distinct numbers observed at least once in the first $n$ extractions, $n \in \mathbb{N}$. Find $E(X_n)$ and $Var(X_n)$. (Tip: $X_1 = m$)
Since the number of extractions will be at least $m$ and at most $k$, given that $X_1 = m$ I tried to evaluate $P(X_n = j + m | X_1 =m)$ for $0\leq j \leq \min\{m, k-m \}$. It turns out that $X_2$ have a hypergeometric distribution but I do not seem to find an answer trying to expand it for $X_n$ and I am really stuck.
Any tips or advices?
Thank you so much!
I think it will be much easier to go about this if we consider the random variables $Y_i,\ i=1,\dots,k$ where $Y_i$ is $1$ if ball number $i$ is drawn at some time in the first $n$ draws and $0$ if it is not. Then $X_n=\displaystyle{\sum_{i=1}^kY_i}$ and it's easy to compute $E(X_n)$ by linearity of expectation. The variance is harder because the $Y_i$ aren't independent, but you can use the formula for the variance of the sum in terms of the covariance.