The US Geological Survey (USGS) monitors occurrences of earthquakes across the world (see, for example, http://earthquake.usgs.gov/earthquakes/search/). Suppose a USGS researcher is interested in the annual number of earthquakes worldwide that are between 6.0 and 6.9 on the Richter scale, and considers a Poisson model with some mean $\mu$ to be appropriate for this variable. The researcher takes a random sample of twenty years for which reliable data exist, and observes the following numbers of counts of earthquakes of interest on those years: \begin{align*} & , \\ & \end{align*} The mean of the above counts is . In performing a Bayesian analysis, the researcher takes as a prior distribution for $\mu$ a density function proportional to $$\left( \mu \right) ^{-1}e^{-\mu }.$$

Part a) Use R to compute a two-sided 95% Bayesian credibility interval for $\mu$, giving your answers to two decimal places.

Lower bound: Upper bound:

Part b) Suppose in the above analysis the researcher chose a Gamma prior distribution, $Ga\left( \alpha ,\beta \right)$, and let $\alpha \rightarrow 0$ and $\beta \rightarrow 0$. In this limit, use R to find the asymptotic two-sided 95% Bayesian credibility interval for $\mu$, giving your answers to two decimal places.

Lower bound: Upper bound: