This is somewhat of a riddle. I am trying to model the probability of filling up party rooms and simulation is not allowed.
I know that the arrival of customers in 1 period of time (1 week) can be modeled with a bayesian negative binomial process ~NB(alpha, beta) where alpha and beta comes from the gamma prior to model the rate parameter. Once a customer arrive, he/she will then be randomly assigned to a specific room with a predetermined probability(example: 25% go to room A, 40% to room B, 35% to room C).
I see this as process as Binomial(X, p) where X ~ NB(alpha, beta). If I have alpha, beta and p and need to calculate the [0.5, 0.95] quantile for the length of time it takes for the number of humans in room A (or B or C) to reach a certain N number of human, I am stuck at how to derive this solution analytically without simulation.
Any thought would be useful.
2026-02-24 06:11:22.1771913482
Predict the probability of rooms being filled from a negative binomial process for arrival
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