1. Simple Metropolis: Normal Precision Gamma. Suppose X : 2 was observed from the population distributed as N (0, FE) and one wishes to estimate the parameter (9. (Here (9 is the reciprocal of the variance 02 and is called the precision para-meter). Suppose the analyst believes that the prior on (9 is Qc(l/2, 1). Using Metropolis algorithm3 approximate the posterior distribution and the Bayes' esti- mator of (9. As the proposal distributiom use gamma Gabon? (9) with parameters a, (3 selected to ensure eicacy of the sampling (this may require some experimenting). (a) Describe the posterior distribution of (9 using the Bayes' estimator and the 97% l-ll'D credible set. (b) Create two plots: one for the posterior density of (9 and one trace plot. For the trace plot, the Xaxis should be the iteration, and the Yaxis should be the observed value of the chain at that iteration. (c) Report the acceptance rate of your proposal distribution. That is, what is the prob- ability that the proposal was accepted when you ran the Metropolis algorithm