Analyze ad targeting expectations and distributions

You manage an advertising slot that serves two audience groups: High-Intent (H) making up 90% of traffic and Low-Intent (L) making up 10%. When an ad is displayed:

• For H, the click-through probability is 0.30; conditional on a click, the conversion probability is 0.40.
• For L, the click-through probability is 0.05; conditional on a click, the conversion probability is 0.10.
• Each conversion brings in $10. Each impression costs $0.002 (equivalent to a $2 CPM). Impressions are assumed independent.

1. Calculate the expected profit for 1,000 impressions under three approaches: (S1) display to all users; (S2) display only to users predicted as High-Intent, given a classifier with 95% precision and 80% recall for distinguishing H from L; (S3) display only when a user's posterior probability of conversion $P(\text{convert})$ is above a cutoff $t$. Determine the threshold $t^*$ that maximizes profit and provide its numerical value given these economic parameters.

2. For strategy S1, find the variance of the conversion count over 1,000 impressions. Present the decomposition using the law of total variance across the H/L mixture. Indicate whether this mixture raises or lowers overdispersion compared to a single Bernoulli distribution with the overall average conversion rate.

3. Your product manager suggests "route every impression to High-Intent users only." Using your earlier findings, list two quantitative advantages and two drawbacks (for instance, effects on reach and how profit responds to precision/recall mistakes).

4. Exponential inter-arrival model: Suppose user sessions follow a Poisson process with rate $\lambda = 0.2$ per minute. a) Give the PDF and CDF for $T \sim \text{Exp}(\lambda)$. Calculate $E[T]$, $Var(T)$, and $P(T > 10)$. b) Let $\bar{T}_n$ denote the sample mean of $n$ independent and identically distributed draws from $\text{Exp}(\lambda)$. State the limit of $\bar{T}_n$ as $n \to \infty$ and the approximate distribution of $\sqrt{n} (\bar{T}_n - 1/\lambda)$ for large $n$ (identify the theorem). Briefly explain how this relates to why large-sample estimates of average time spent become stable in dashboards.

5. If actual traffic is a 90/10 H/L mixture with differing click propensities, does the inter-click-time distribution remain exponential? If not, describe the resulting distribution family in qualitative terms and name one diagnostic plot you would use to identify the mixture.

Overview: This question assesses a data scientist's skills in probabilistic modeling, decision-making based on expected value, variance decomposition for mixture distributions, and analysis of Poisson/exponential arrival processes, all within the Statistics & Math domain.