The term paper: Problem 2.

Updated April 3rd; today's version of the Schweder compendium now has exercise 13 back to number "13". In any case, the clickable link in the problem set leads to the appropriate problem

There has been a question on the interpretation of problem 2, which is exercise 13 in the April 3 version of Schweder's compendium.

First part, the result concerning the posterior distribution is found in the problem text -- and it will give you the g function in Schweder p. 35, with the form as in the solution.
So the answer is given, but you are supposed to show this in sufficient detail. (No numerics here.)

Second part, the numerical exercise bit, the MCMC is outlined in Scwheder pp. 35--36 (p. 35 in the old version of the compendium): 

• You already have g (modulo a constant), and q is given in the solution (where φ is the standard Gaussian density function) together with ρ
• Then you
  (1) Start at some arbitrary point s in the state space; put T₁=s.
  (2) When you have drawn T_n, draw U_n as described on p. 36 (p. 35). You can even use a spreadsheet
        (hint: you only need to draw one standard normal variable, add T_n -- and what would you do to keep it positive? What is done in the solution?)
  (3) The next state is then given by formula (14) -- read the g(U_n)>g(T_n) criterion in the subsequent paragraph.
  (4) Run this recursively for a sufficient number n of iterations. Store T_N.
  (5) Repeat (1) -- (4) a sufficient number of times. For each repetition, store T_N.
  (6) Plot the distribution of the T_N against the gamma. A qq-plot is preferred, but for the purposes of the term paper, a histogram together with the would also do.
  
  (If in (5) you are using a spreadsheet, then just use one column for each repetition (1) -- (4), and copy formulas (just make sure you get independent randomness).)