3. Casting defects. The number of defects in an iron casting can be assumed to follow a Poisson distribution with mean A. 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 0 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 A (generate 100,000 samples and use 1,000 samples as burn-in) and answer the following: 1. Plot the posterior density of 0. 2. Find the posterior mean of 0. 3. Find 95% equitailed credible interval of 0. Note: the density of a Gamma(a, b) is given by b"/T(a)ro le " and the density of Exp(1) is given by de Ar