Non-Uniform Coupon Collector Problem via Inclusion-Exclusion

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Inclusion-Exclusion / Continuous Embedding.

Your Task
Suppose there are $m$ distinct coupons, where coupon $i$ has probability $p_i > 0$ of being drawn ($\sum_{i=1}^m p_i = 1$). Let $T$ be the number of draws until all $m$ coupons have been observed at least once. Derive the exact expression for $\mathbb{E}[T]$ using:

1. The Principle of Inclusion-Exclusion.
2. Embedding into continuous time via independent Poisson processes.

Constraints

• Samples are independent and identically distributed on their respective support.
• Boundary conditions are defined strictly within standard real intervals.