Model user-level ad impression allocation

Suppose there are $X$ distinct users and $Y$ ad impressions. Every impression is independently assigned to a uniformly random user, and impressions are assigned with replacement. For a fixed user $u$: (a) Derive $\mathbb{E}[\text{Impressions}_u]$ and $\mathrm{Var}(\text{Impressions}_u)$. (b) Compute $P(u \text{ sees at least one impression})$. (c) Derive $\mathbb{E}[\# \text{ of users who see at least one impression}]$. (d) When $X$ is large and $Y/X$ is small, provide the first-order Taylor approximation and an exponential (Poisson) approximation for part (b), and indicate when each approximation is accurate. (e) State the exact assumptions that make a Binomial model appropriate here, and when a Poisson approximation is suitable.

Overview: This question tests probability and statistical modeling, including binomial occupancy, expectations and variances, event probability calculations, and Taylor and Poisson approximations for count data; it belongs to the Statistics & Math domain.