4. Now suppose that the gram of radioactive material emits oz-particles according to a Poisson process {N (t) : t Z 0} at the specic rate of A = 3.2 particles per second. This exercise will walk you through an algorithm for generating samples of the arrival times of the particles in the rst two seconds. (a) (b) What is the distribution of N (2), the number of particles that will arrive in the rst two seconds? You don't need to explain why your answer is correct - just name the distribution. Next, we need to generate a sample of N (2), which can be done by implementing Hastings-Metropolis MCMC. Choose a starting point for X0 6 {0, 1, 2, ...}, and select a proposed move Y for X1. A good choice might be a Symmetric random walk on {0, 1, 2, ...} (but you need to be careful at the endpoint 0). For each sample of the proposed move, you will need to implement an accept/reject mechanism following the Hastings- Metropolis formula. Use this procedure (you should manually sample coin ips and uniform samples from [0,1] as needed - feel free to deter- mine ways to do the latter by computer, or simply pretend!) to generate X1.X2,X3, and X4, and set N(2) = X4 as your sample for N(2). Your submission should involve you walking through each step of the algorithm as you generate the samples, and accept/reject, etc. Note that the choice of n = 4 was arbitrary, in practice we would do this by computer and generate Xn for n > 4 as our sample for N (2) If you are inclined to solve this exercise by writing code (which would allow you to set N (2) = X\" for n large) you are certainly encouraged to do that, but in that case please submit a copy of your code. Suppose N (2) = k was the sample you generated in 4(b) above (if you were unable to complete exercise 4(b), feel free to pick some reasonable value for N (2) at this stage). Now generate samples of the arrival times for the is emitted aparticles in the rst 2 seconds. As above, your sub- mission should involve you walking through the details of the procedure you are implementing to generate the samples