Ads Profit, Variance Decomposition, and Exponential Timing

Context: You run an ad slot with two user segments. On each eligible page view (impression opportunity), you may choose to show an ad. If shown, you incur a cost and may earn revenue from a conversion.

Segments and probabilities:

• High-Intent (H): 90% of traffic
  • P(click | shown) = 0.30; P(convert | click) = 0.40 → P(convert | shown, H) = 0.30 × 0.40 = 0.12
• Low-Intent (L): 10% of traffic
  • P(click | shown) = 0.05; P(convert | click) = 0.10 → P(convert | shown, L) = 0.05 × 0.10 = 0.005
• Economics: revenue per conversion R = $10; cost per impression c =$ 0.002 ($2 CPM)
• Assume independence across impression opportunities.

Tasks:

1. Expected profit per 1,000 eligible opportunities for three strategies:
   • S1: show to everyone
   • S2: show only to predicted-High users; your classifier has precision = 95% and recall = 80% for H vs. L
   • S3: show only when a user’s posterior P(convert) ≥ threshold t. Derive the profit-maximizing threshold t* and give its numeric value under these economics.
2. Under S1, compute the variance of the number of conversions per 1,000 impressions. Use the law of total variance to decompose across the H/L mixture, and state whether the mixture increases or decreases overdispersion relative to a single Bernoulli with the average conversion rate.
3. Your PM proposes: "send all impressions to High-Intent only." Give two quantitative pros and two cons using your results (e.g., reach, profit sensitivity to precision/recall).
4. Exponential inter-arrival model. Sessions arrive as a Poisson process with rate λ = 0.2 per minute. a) Write the PDF and CDF of T ~ Exp(λ). Compute E[T], Var(T), and P(T > 10). b) Let T̄_n be the sample mean of n IID Exp(λ). State the limit of T̄_n as n → ∞ and the approximate distribution of √n (T̄_n − 1/λ) for large n (name the theorem). Explain why large-sample averages stabilize in dashboards.
5. If real traffic is a 90/10 H/L mixture with different click propensities, is the inter-click-time distribution exponential? If not, name the resulting family qualitatively and one diagnostic you would plot to detect the mixture.