Let the observed data be y = (6, 4, 9, 2, 0, 3), a random sample from the Poisson distribution with mean A, where A> 0 is unknown. Suppose that we assume a Gamma(1, 1) prior distribution for A. The posterior density, p(A | y), for λ is Gamma (1+ S, 1 + n), where S = ₁ and n = 6. Suppose that you want to construct a symmetric Metropolis-Hastings on the log-scale to generate a sample from this posterior distribution by using a normal proposal distribution with standard deviation b = 0.2. (a) Write down the steps in this symmetric Metropolis-Hastings (on the log-scale) to simulate realisations from the posterior density p(x|y). (b) Implement the algorithm in R. and plot the observations as a function of the iter- ations. Use M = 5000 for the number of iterations. (c) To assess the accuracy compare the empirical distribution of the sample with the exact posterior density, Gamma(1 + S, 1 + n). (d) Rerun the algorithm in R using a smaller b = 0.01 and a larger b = 20. What are the effects on the behaviour of the algorithm of making b smaller? What are the