Compute Poisson supply–demand match probability

Q: Compute Poisson supply–demand match probability

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Question

In a city-day of a two-sided marketplace, customer demand D ~ Poisson(8) and supplier capacity S ~ Poisson(6), assumed independent. (a) Compute P(S >= D). Express it exactly using the Skellam distribution for (S − D) and provide a numerical approximation to 3 decimals. (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 helpful, or propose a simulation strategy with error bounds. (c) On weekends, demand rises to D ~ Poisson(9). By how much must capacity increase (i.e., find the smallest nonnegative δ such that S' ~ Poisson(6 + δ)) to achieve P(S' >= D) ≥ 0.90? Provide the smallest integer δ that satisfies the constraint, and justify your method (closed-form, normal/Skellam approximation, or simulation with confidence).

Compute Poisson supply–demand match probability

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