Compute Poisson supply–demand match probability

On a city-day in a two-sided marketplace, customer demand is $D \sim \mathrm{Poisson}(8)$ and supplier capacity is $S \sim \mathrm{Poisson}(6)$, with the two assumed independent.

(a) Compute $P(S \ge D)$. Give the exact form using the Skellam distribution for $S - D$, and provide a numerical approximation to 3 decimal places.

(b) Derive $E[\min(S, D)]$ and the expected fill rate $E[\min(S, D)] / E[D]$. You may use identities involving $E[|S - D|]$ if that helps, or describe a simulation strategy with error bounds.

(c) On weekends, demand rises to $D \sim \mathrm{Poisson}(9)$. By how much must capacity increase—that is, find the smallest nonnegative $\delta$ such that $S' \sim \mathrm{Poisson}(6 + \delta)$—to achieve $P(S' \ge D) \ge 0.90$? State the smallest integer $\delta$ that satisfies the constraint and explain your method (closed-form, normal/Skellam approximation, or simulation with confidence).

This problem tests understanding of probability distributions and stochastic modeling—specifically Poisson processes, the Skellam distribution for differences, expectation of minima, and probabilistic capacity planning within the Statistics & Math domain.

This question is drawn from a Lyft Data Scientist interview experience.