Expected impressions per user under random assignment

Q: Expected impressions per user under random assignment

This question evaluates understanding of probability and statistical modeling, focusing on expected values, occupancy/assignment distributions, and Poisson approximations in the context of randomized ad impression allocation, placed in the Analytics & Experimentation domain for data science roles.

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Question

Random Assignment of Ad Impressions Across Users

Setup

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; multiple impressions can be delivered to the same user.

Questions

For an arbitrary user i, what is the expected number of impressions they receive?
What is the probability that user i receives at least one impression?
Provide both the exact expressions and the large-scale Poisson approximations.

Expected impressions per user under random assignment

Random Assignment of Ad Impressions Across Users

Setup

Questions

Solution

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