Answer the following:

Implement a Metropolish-Hastings algorithm to sample from the Rayleigh distribution with scale parameter o. The Rayleigh distribution is used to model wind speeds, lifetimes of certain objects, and service times. The Rayleigh density is given by TT ( 20 ) = 02 ex /(20"), for x > 0,0> 0. The mode of this distribution is o, the mean is ov7/2, and the variance is o2(4 - 7)/2. For the proposal distribution, use q(x, y) ~ x2 ( df = x), the chi-square distribution with a degrees of freedom. The proposal density is q(x, y) = y"/2-le-y/2 2*/2T(2 for y 2 0. (a) Write down the framework of your Metropolis-Hastings algorithm for this problem. (b) Let o = 4. Run the Metropolis-Hastings algorithm for 6000 iterations and make a trace plot (or time plot) of the Markov chain. Hint: You may use the dchisq () and rchisq () functions for this question. (c) Use the samples generated in the last 5000 iterations to estimate the mean of Rayleigh(o = 4): E. (X)