Two Quiet Days at the Help Desk
A help desk receives urgent tickets as a Poisson process with an unknown rate $\lambda$ per day. Your prior for $\lambda$ is Gamma with shape 2 and rate 1 (mean 2). You then observe 4 tickets in total over 2 days. What is the posterior predictive probability that the next 2 days together produce no urgent tickets at all? Give your answer as a reduced fraction.
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Source: Original problem written for AxiomIQ (standard Gamma-Poisson posterior predictive, i.e. negative binomial). Statement written for AxiomIQ.