Expected impressions per user under random assignment
Quick Overview
Evaluates random assignment of ad impressions across users using binomial and Poisson models. Strong answers compute expected impressions, at-least-one exposure probability, exact expressions, and large-scale approximations.
Expected impressions per user under random assignment
Company: Meta
Role: Data Scientist
Category: Analytics & Experimentation
Difficulty: easy
Interview Round: Onsite
Scenario: An experiment drops Y ad impressions randomly across X users. Compute per‑user expectation and the probability an individual sees at least one impression.
Question 1: With Y impressions randomly assigned to X users, compute expected impressions per user and probability of at least one. (Hint: Poisson approximation, independence)
Overview: Evaluates random assignment of ad impressions across users using binomial and Poisson models. Strong answers compute expected impressions, at-least-one exposure probability, exact expressions, and large-scale approximations.
In an A/B experiment, Y ad impressions are served uniformly at random across X distinct users. Each impression is independently assigned to one of the X users, and multiple impressions can be delivered to the same user.
Constraints & Assumptions
Each impression independently chooses one user uniformly at random.
X and Y are positive integers.
Focus on one arbitrary user i.
Provide exact expressions and large-scale Poisson approximations.
Clarifying Questions to Ask Guidance
Is assignment truly independent across impressions?
Can users receive multiple impressions?
Are all X users equally eligible for every impression?
Is the goal user-level exposure probability or load distribution?
Part 1 - Expected Impressions
For an arbitrary user i, what is the expected number of impressions they receive?
What This Part Should Cover Guidance
Model the count as Binomial(Y, 1/X).
Use linearity of expectation to get Y/X.
Optionally state the variance for intuition.
Part 2 - Probability of At Least One Impression
What is the probability that user i receives at least one impression?
What This Part Should Cover Guidance
Use the complement of receiving zero impressions.
Provide the exact expression 1 - (1 - 1/X)^Y.
Part 3 - Poisson Approximation
Provide the large-scale approximation.
What This Part Should Cover Guidance
Let lambda = Y/X.
Approximate the Binomial count with Poisson(lambda) when X is large and 1/X is small.
Approximate at-least-one probability as 1 - exp(-lambda).
Follow-up Questions Guidance
What is the expected number of users who receive at least one impression?
How would the result change if assignment probabilities differ by user?