Question
Consider a set $\mathcal{U}$ of $n$ distinct objects. At each time $t\geq 0$, we denote by $V(t)$ the number of visited objects. Assume $V(0)=0.2 n$, and for any $t>0$, $V(t)=V(t-1)+0.65\sqrt{n}-X_{t}$, where $X_t\sim Hypergeometric(n,V(t-1),\sqrt{n})$.
Compute $Var(V(t))$.
What I have done
I can compute the expected value of $V(t)$ (see below). However, I need help to compute $Var(V(t))$.
Computing $E[V(t)]$
$E[V(t)] = E[V(t-1)] + 0.65\sqrt{n} - V(t-1)/\sqrt{n}$. Let's take an expectation from both sides: $E[V(t)] = E[V(t-1)] + 0.65\sqrt{n} - E[V(t-1)]/\sqrt{n}$. Hence, we have: $E[V(t)] = E[V(t-1)](1-1/\sqrt{n}) + 0.65\sqrt{n}$. Now, we can easily solve this recursive relation.