1. Parametric Estimation using the Likelihood Function:
The maximum likelihood technique estimates a parameter set , by maximizing :

(1)

a data set of N vectors.

In practice, we define an error function, E, to be minimized:

(2)

For the Gaussian distribution in one dimension:

(3)

find the values of the mean and variance that minimize the error function (equation 2).
 

2. Go over section 2.6 in Bishop on Mixture models.

2.1 Derive: and (equations 2.79 and 2.80).

2.2 Show and 
(equations 2.85 and 2.86).

2.3 Explain in your own words the intuition behind the update equations:

(equations 2.96-2.98).