In the coupon collector problem, a collector gets a random coupon from $\{1,\ldots,n\}$ at every time instance.
The original formulation usually deals with bounds on the time it takes the collector to collect all (distinct) coupons.
What if we define the opposite problem:
- For a time $t\in\mathbb N$ let $C_t$ denote the number of distinct coupons within the first $t$ coupons.
By looking at some coupon $i\in\{1,\ldots,n\}$, the probability that it'll be collected is $1-(1-1/n)^t$, and thus we have $\mathbb E[C_t] = n\cdot (1-(1-1/n)^t)$.
I'm interested in a lower bound $x_t$ for $C_t$; specifically,
For some $\delta>0$, how can we find $x_t$ such that $$\Pr[C_t<x_t]\le \delta\quad{}\mbox{?}$$