Radioactive decays per second, Gamma-Poisson

Question

You model radioactive decays as a Poisson process with unknown rate $\lambda$ per second. Your prior is $\text{Gamma}(\text{shape } 1, \text{ rate } 0.5)$ , where the rate parameter counts seconds already observed. You then run the detector for 12 seconds and count 30 decays.

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

Accepted Answer

Intuition first. The exposure is 12 seconds, and the prior's rate parameter (0.5) can even be a fraction of a second of pretend observation. The Gamma-Poisson rule is the same: add total decays to the shape and total seconds of exposure to the rate parameter. The rule. With prior 	ext{Gamma}(\alpha, \beta), total events \sum xi, and total exposure t seconds: 	ext{posterior} = 	ext{Gamma}\Big(\alpha + \sum xi,\ \ \beta + t\Big). Plug in. Here \alpha = 1, \beta = 0.5, events \sum xi = 30, exposure t = 12 seconds: 	ext{posterior} = 	ext{Gamma}(1 + 30,\ 0.5 + 12) = 	ext{Gamma}(31,\ 12.5). P…

Radioactive decays per second, Gamma-Poisson

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