The expected value part is straightforward, just Y divided by X.
Start by clarifying the assumptions: each impression is independently assigned to a uniformly random user, so the number of impressions per user follows a Binomial distribution. Then compute the expected value and the probability of at least one impression using the complement rule, and discuss practical implications for ad delivery and experimentation.
Pro tip: Mention that in real systems, impressions are not perfectly independent due to user targeting and frequency capping, so the Binomial model is a simplification. Acknowledging this shows you understand the gap between theory and production.
Confirm that each impression is independently and uniformly assigned to one of X users, and that Y is fixed. This leads to a Binomial model for each user's impression count.
Let I_i be the number of impressions user i receives. Then I_i ~ Binomial(Y, 1/X).
By linearity of expectation, E[I_i] = Y * (1/X) = Y/X. This is the average number of impressions per user.
Use the complement: P(I_i >= 1) = 1 - P(I_i = 0) = 1 - (1 - 1/X)^Y. For large X and Y, approximate with 1 - e^{-Y/X}.
Relate to ad delivery: if Y/X is small, many users see zero ads; if large, users see multiple ads. Mention that in practice, targeting and frequency capping alter these probabilities.
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