Estimate bots and CI from DAU spike
Quick Overview
This question evaluates proficiency in mixture modeling for anomaly detection, parametric and nonparametric inference for mean differences, handling overdispersed count data, and bootstrap resampling under heavy-tailed distributions.
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Oct 13, 2025, 9:49 PM
Data Scientist
Onsite
Statistics & Math
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Mixture Spike and Mean-Difference Inference for Daily Comments
Context
A product has DAU (daily active users) = 2,000,000. On day T, total comments increased by 20%. Historically, the per-user mean is 5.5 comments; on day T, the mean is 6.6 (= 1.2 × 5.5). You suspect a small fraction of bot accounts that each posted around 500 comments on day T.
Tasks
- Two-component mixture model: assume humans remained at 5.5 comments/user and bots posted 500 comments/user. Estimate the number and fraction of bot accounts on day T. Show formulas and a numeric result.
- Re-estimate if the human mean drifted up to 5.8 on day T (bots still at 500).
- From day-T DAU, two independent simple random samples (SRS) of size 1% are drawn (Sample A and Sample B). Compute a 95% confidence interval (CI) for the difference of sample means, mean(A) − mean(B): a) assuming per-user comments are Poisson with mean 6.6. b) assuming Negative Binomial with mean 6.6 and dispersion k = 2 (use Var = μ + μ²/k). State and justify approximations (e.g., CLT). You may comment on finite population correction (FPC) if relevant.
- If the true distribution is heavy-tailed with ~1% of humans having >50 comments, discuss when the normal approximation can break and outline a nonparametric bootstrap procedure to obtain a CI for mean(A) − mean(B).
- Briefly critique assuming normality and propose a quick diagnostic using only sample aggregates (n, mean, variance).
Solution
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