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.