Homework 12, Due Monday Oct 26, 2009.
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This problem involves implementation of the EM algorithm 
in a "mixture" setting with Poisson data, where the 
Poisson parameter lambda is drawn with specified 
probabilities from a set {4*mu,5*mu,7*mu} with three 
values.

(a) Generate 1000 random counts X1,...,X1000
from the discrete probability mass function

P(X=k) = p(k) = (mu^k/k!)*( .2*(4^k*exp(-4*mu)) + 
      .5*(5^k*exp(-5*mu)) + .3*(7^k*exp(-7*mu)) )

where you choose a fixed value for mu (say mu=1.2, or mu = 0.7).
This is a "mixture" with weights .2, .5, .3 of respective 
Poisson(4*mu), Poisson(5*mu), and Poisson(7*mu) random variates.

(b) Find the maximum likelihood estimate of mu from your data 
 in two ways:  
     (1) by a direct log-likelihood maximimization using one of 
       the R-supplied minimization functions "nlm" or "optim".

     (2) By the EM algorithm, which you should set up and code 
       using the ideas presented in class.

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[The old HW listed as HW12 has now been changed to HW13 and 
 will be due Friday, Oct. 30.]