Gamma-Poisson with unequal exposure time

Question

You model machine breakdowns as a Poisson process with unknown rate $\lambda$ measured per 100 hours. Your prior is $\text{Gamma}(\text{shape } 2, \text{ rate } 1)$ , where the rate parameter counts units of 100 hours already observed. You then watch the machine for 350 hours (that is 3.5 units of 100 hours) and record 6 breakdowns.

Using Bayesian updating, what is the posterior mean estimate of the breakdown rate $\lambda$ (per 100 hours)?

Accepted Answer

Intuition first. This is the usual Gamma-Poisson update with one wrinkle: the amount observed is not a whole number of periods but 3.5 units of exposure time. The rule still holds if you add total events to the shape and total exposure (in the same units as the rate) to the rate parameter. Think of the rate parameter as "units of time already watched." The rule. With prior 	ext{Gamma}(\alpha, \beta), total events \sum xi, and total exposure t (in the rate's time units): 	ext{posterior} = 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + t\Big). In words: new shape = old shape + events, new rate…

Gamma-Poisson with unequal exposure time

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