Consider a univariate normal model with mean and variance . Suppose we use a Beta(2,2) prior for (somehow we know is between zero and one) and a log-normal(1,10) prior for (recall that if a random variable X is log-normal(m, v) then logX is N(m, v)). Assume a priori that and are independent. Implement a Metropolis-Hastings algorithm to evaluate the posterior distribution of and . Remember that you have to jointly accept or reject and . Also compute the posterior probability that is bigger than 0.5.

Here are the data:

2.3656491 2.4952035 1.0837817 0.7586751 0.8780483 1.2765341 1.4598699 0.1801679 -1.0093589 1.4870201 -0.1193149 0.2578262