Answer the following:

1. The Poisson distribution with mean parameter A has probability mass function ihAm 7r(a:)=e I , for$=0,1,2,.... 3:. Consider the proposal distribution :1: + 1, with probability % .1: 1, with probability % H.321, y: ay): { If {B = 0, y = :1: + 1, with probability 1. We want to implement a Metropolis-Hastings algorithm to sample from the Poisson()\\) distribution. (a) Write down the framework of your MetropolisHastings algorithm for this problem. (b) Let A = 6. Run the MetropolisHastings algorithm for 2000 iterations and make a trace plot (or time plot) of the Markov chain. (c) Calculate the mean and plot the histogram of the last 1000 samples