Coupon Collector Problem and Harmonic Expectation

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Coupon Collector / Harmonic Numbers.

Your Task
There are $n$ unique coupon types. Each box of cereal contains one coupon chosen uniformly at random with replacement. Let $T$ be the number of boxes purchased until at least one coupon of each of the $n$ types is collected.
(1) Derive the exact expected value $\mathbb{E}[T]$ in terms of harmonic numbers $H_n = \sum_{k=1}^n \frac{1}{k}$.
(2) Derive the exact variance $\text{Var}(T)$.
(3) Compute the asymptotic behavior as $n \to \infty$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.