HW 19 Stat 705  Fall 2015

Assigned Wednesday 12/9/15 

DUE Thursday 12/17/15, worth 15 points

(1)  A pair of random variables (X,Y) is assumed to 
have joint density such that  

given Y ,      X ~ Expon(lambda)    with rate lambda = 1+2*Y

given X ,      Y ~ Beta(1+2X, 3+X^2)

Write an R function to simulate random pairs (X_i,Y_i) from 
this joint density (not necessarily independent across i), 
and use that function to find (either through a picture, a 
density function, or some other representation) the density 
function of X.


(2) Generate a small dataset according to the commands

set.seed(7793)
xv = runif(80)
yv = 1.5*xv + 0.8*rnorm(80)

Use these data to simulate the posterior density of  b given (xv,yv)

in the model:

      yv[i] = b* xv[i] + sig*eps[i]  with  eps i.i.d. N(0,1), i=1,..,80

where (b, tau) = (b, 1/sig^2) are viewed as unknown statistical 
parameters with prior density   b ~ N(0, 10^2), tau ~ Gamma(0.5,.05).

To solve this problem you must implement the Gibbs Sampler directly 
in R , and hand in your code for it, rather than use any 
pre-programmed package.

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For both parts of this HW set: show whatever checks you can to establish 
that your R code does what it is supposed to, i.e. that the simulated
(Gibbs-Sampled) sequences have the desired properties.