We return to the scenario of the previous question, but this time we will consider a Bayesian inference. Suppose now that there were students, and the total number of errors they counted in issues of The Guard was , the data being \begin{align*} \\ \end{align*}

Part a) Evaluate the likelihood function (including terms not involving $\mu$) for the data when the mean number of errors per issue is . Give the natural log of the likelihood to two decimal places.


Part b) In performing a Bayesian analysis, you take as a prior distribution for $\mu$ a density function proportional to \begin{align*} \mu^{-1}e^{-\mu }. \end{align*} Find the prior probability that the mean number of errors per issue is less than five, giving your answer to four decimal places.


Part c) Find your posterior probability that $\mu <4$, giving your answer to four decimal places.


Part d) Suppose we are interested in testing $H_{0}:\mu \geq 5$ against the alternative $H_{a}:\mu <5$. Find the posterior probability for the null hypothesis, giving your answer to four decimal places.


Part e) Find the posterior probability for the alternative hypothesis, giving your answer to four decimal places.