Homework 17, Due Friday Nov 20, 2009.
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This exercise implements an example (Example 3, p. 172)
covered in the paper, "Explaining the Gibbs Sampler", by
Casella and George, American Statistician (1992) 46, 167-174.

(a) Generate (and time) a direct simulation of 1000 iid 
nonnegative-integer valued random variables 
X_1, X_2, ..., X_{1000} from the marginal distribution 
constructed in the Homework Handout  HW17Exmp.pdf.
(Use the idea given at the bottom of p.1 to top of p.2 
of the Handout.

(b) Write code to generate a Gibbs Sampler simulation of
variables X_1, X_2, ..., from a Markov Chain with equilibrium 
distribution the same as the marginal X distributon f(x)
from HW17Exmp.pdf.

Create some graphical exhibit (e.g. a histogram or appropriate 
Table) to show that you have simulated random variables X_i 
with the same (marginal) distribution in both parts (a), (b).


(c) To do a fair comparison of the running time of direct and 
Gibbs-Sampler simulation of iid X's, you need to figure out how 
large the integers K and  T should be so that the variables 
X_K, X_{K+T}, X_{K+2T}, ... X_{K+999T}  are all approximately 
independent with the marginal probability mass function  f(x).