Model user-level ad impression allocation
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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Oct 13, 2025, 9:49 PM
Data Scientist
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Statistics & Math
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Random Assignment of Ad Impressions to Users
Context
- There are X distinct users and Y ad impressions (X ≥ 1, Y ≥ 0 integers).
- Each impression is independently assigned, uniformly at random, to one of the X users (with replacement across impressions).
- Fix a particular user u.
Tasks
(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, provide:
- the first-order Taylor approximation to part (b), 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.
Solution
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