Model user-level ad impression allocation
Company: Meta
Role: Data Scientist
Category: Statistics & Math
Difficulty: medium
Interview Round: Onsite
You have X distinct users and Y ad impressions. Each impression is independently assigned uniformly at random to one user (with replacement across impressions). For a fixed user u: (a) Derive E[Impressions_u] and Var(Impressions_u]. (b) Compute P(u sees at least one impression). (c) Derive E[# of users who see at least one impression]. (d) For large X with Y/X small, give the first-order Taylor approximation and an exponential (Poisson) approximation to part (b), and state when each is accurate. (e) State the precise assumptions that justify using a Binomial model here, and when a Poisson approximation is appropriate.
Overview: This question evaluates probability and statistical modeling skills, specifically binomial occupancy, expectations and variances, event probability calculations, and the use of Taylor and Poisson approximations in count data; it falls under the Statistics & Math domain.
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