%Andrew McCallum
%Problem Set 11
%Nov. 12, 2007
%Question 2

close all;
clc;
clear;

%Increase the display precision of my variables.
format('long','g');

%2.a.
randn('seed', 1234);
% Normal mean = 1 variance = 6
mu = 1;
sig2 = 6;
sig = sqrt(sig2);
%Matrix dimensions
m = 100;
n = 1;
x = normrnd(mu,sig,m,n);

%2.b.
%MatLab divides by (m-1) (sample size minus 1) our closed form MLE estimator 
%divides by m, hence the difference.
sig22b = var(x);
xbar = mean(x);
sig22bclosed = sum((x - xbar).^2)/m;

%2.c.
%Get the MLE estimate via numerical methods.
%Because theta is now a vector of parameters we need to specify two initial
%values for x0.

x0 = [0.001, 8.75342];
theta2c = fminsearch(@(theta) lnnormlike(theta,x), x0) 

%2.d.
%Solve the system of nonlinear equations for theta, a 2 X 1 vector
theta2d = fsolve(@(theta) lnnormfoc(theta,x), x0)