3. Casting defects. The number of defects in an iron casting can be assumed to follow a Poisson distribution with mean . A quality engineer inspected nine castings and observed the following number of defects in them: 0,2,2,3,3,1,2,1,1. Assume that has a prior distribution Gamma(2,b), where the hyperparameter b is assumed to have a distribution Exp(1). Use Gibbs sampling to sample from the posterior distribution of (generate 100,000 samples and use 1,000 samples as burn-in) and answer the following: 1. Plot the posterior density of . 2. Find the posterior mean of . 3. Find 95% equitailed credible interval of . Note: the density of a Gamma(a,b) is given by ba/(a)xa1ebx and the density of Exp() is given by ex.