Metropolis-Hastings Algorithm

1. Suppose that the number of children per household in the U.S. has the following distribution

$$\begin{array}{rlrlrlrlrlrlr}x& & 0& & 1& & 2& & 3& & 4& & 5\\ \text{P}(X=x)& & 0.15& & 0.18& & 0.35& & 0.20& & 0.08& & 0.04\end{array}$$

1. Describe in detail how you could implement by hand a Metropolis-Hastings algorithm to simulate from this distribution, using a random walk to neighboring states as the proposal chain.
2. Write a few lines of code to implement and run your algorithm, and summarize the approximate distribution. Does your algorithm seem to work?
3. Specify the transition matrix for the M-H algorithm in the previous part.
4. Find the stationary distribution of the M-H chain. Is it the target distribution?

2. The standard double exponential distribution has a pdf which satisfies $$f(x)\propto e^{-|x|},-\mathrm{\infty }<x<\mathrm{\infty }$$

1. Describe in detail a Metropolis-Hastings algorithm for simulating from this distribution.
2. Write a few lines of code to implement the algorithm and run it. Does the algorithm seem to produce values from the target distribution?