Coupon Collector's Problem via Geometric Waiting Times

A cereal box contains one of $n$ distinct coupon types, each equally likely. You buy boxes one at a time, independently.

Let $T$ be the number of boxes needed to collect all $n$ types.

(a) Define $T_i$ as the number of additional boxes needed to go from $i-1$ distinct types to $i$ distinct types. What is the distribution of $T_i$? State its parameter.

(b) Express $T$ in terms of $T_1, T_2, \ldots, T_n$ and use linearity of expectation to derive $E[T]$.

(c) Show that $E[T] = n H_n$ where $H_n = \sum_{k=1}^{n} 1/k$ is the $n$-th harmonic number.

(d) Compute $E[T]$ for $n = 10$. How many boxes on average?

(e) Derive $\text{Var}(T)$ using the independence of $T_1, \ldots, T_n$.