Customer arrivals over part of an hour, Gamma-Poisson

Question

You model customer arrivals as a Poisson process with unknown rate $\lambda$ per hour. Your prior is $\text{Gamma}(\text{shape } 5, \text{ rate } 1)$ , where the rate parameter counts hours already observed. You then watch the entrance for 2.5 hours and count 20 arrivals.

Using Bayesian updating, what is the posterior mean estimate of the arrival rate $\lambda$ (per hour)?

Accepted Answer

Intuition first. The exposure is a fractional number of hours, 2.5, but the Gamma-Poisson rule handles that directly: add total arrivals to the shape and total hours of exposure to the rate parameter, which stands for "hours already watched." The rule. With prior 	ext{Gamma}(\alpha, \beta), total events \sum xi, and total exposure t hours: 	ext{posterior} = 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + t\Big). Plug in. Here \alpha = 5, \beta = 1, events \sum xi = 20, exposure t = 2.5 hours: 	ext{posterior} = 	ext{Gamma}(5 + 20,\ 1 + 2.5) = 	ext{Gamma}(25,\ 3.5). Posterior mean = shape ov…

Customer arrivals over part of an hour, Gamma-Poisson

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