Updating an event rate

Idea. The Gamma prior is conjugate to the Poisson likelihood: the posterior is again a Gamma, with the total count added to the shape and the number of windows added to the rate.

Step 1: the posterior. $$\pi (\lambda \mid \text{data})\propto \lambda e^{-\lambda }\cdot {\lambda }^{684}{e}^{-72\lambda }={\lambda }^{685}{e}^{-73\lambda },$$ a Gamma distribution with shape $2+684=686$ and rate $1+72=73$.

Step 2: its mean. $$\mathbb{E}[\lambda \mid \text{data}]=\frac{686}{73}\approx \mathrm{9.40.}$$

Interpretation. The posterior mean is a weighted average of the prior mean 2 and the sample mean $\frac{684}{72}=9.5$, with weights $\frac{1}{73}$ and $\frac{72}{73}$: $$\frac{1}{73}\cdot 2+\frac{72}{73}\cdot 9.5=\frac{686}{73}.$$ The prior counts like one window's worth of data, so 72 windows swamp it.