(Solved): The Poisson distribution with mean parameter has probability mass function (x)=x!ex, ...

The Poisson distribution with mean parameter $?$ has probability mass function $?(x)=\frac{e^{??}{?}^x}{x!},\text{ for }x=0,1,2,\dots$ Consider the proposal distribution $q(x,y):?\begin{array}{ll}\text{ If }x?1,y=\{\begin{array}{ll}x+1,& \text{ with probability }\frac{1}{2}\\ x?1,& \text{ with probability }\frac{1}{2}\end{array}& \\ \text{ If }x=0,y=x+1,& \text{ with probability }1\end{array}$ We want to implement a Metropolis-Hastings algorithm to sample from the Poisson $(?)$ distribution. (a) Write down the framework of your Metropolis-Hastings algorithm for this problem. (b) Let $?=6$. Run the Metropolis-Hastings 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.