Cutting Sample Size with CUPED

Your primary metric is 14-day revenue per user, with standard deviation about $80 and mean $20. The MDE is $1 per user. Pre-experiment 14-day revenue for the same users is available and correlates with the in-experiment metric at \rho = 0.6.

1. Estimate the per-arm sample size without CUPED (use \alpha=0.05, power 0.8).
2. Write the CUPED adjustment and estimate the reduced sample size.
3. Why does CUPED not bias the treatment effect? What would go wrong if you used a covariate measured during the experiment instead?