Homework 1

1. Implement the EM algorithm for estimating the means of a mixture of two
Gaussians, as discuss in the first lecture (and also in section 6.12 of the
Machine Learning text.).  In particular, write a function that generates data
according to a mixture of two Gaussians, they apply your EM algorithm to see
how well it estimates the means.  Try varying the sample size and seeing how
this impacts both accuracy of the estimates and convergence rate.

**NOTE: if you use the Machine Learning text, note there is a bug in equation
(6.31) which is repeated as an unnumbered equation on page 193.  The 1/m term
in this equation should instead be 1/(sum_(i=1 to m) E[z_ij]) ***  (see
www.cs.cmu.edu/~tom/mlbook.html for the full errata for the book.)

2. Write out the EM algorithm (i.e., the Q function) for the problem described
in the paper distributed in class monday Feb 1.