%Problem Set 7 Due Wednesday Oct. 17, 2007

close all; %closes all figures
clc; %clears the screen
clear; %clears the memory

%set the normal and unit density seeds
randn('seed',1234);
rand('seed',0);

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%1. a.
%m x n matrix
m = 100;
n = 1;

%mean and variance
mu = 2;
sigma2 = 5;

%generate the random draws
x = mu + sqrt(sigma2)*randn(m,n);

%plot the vector
%zero line
z = zeros(m,n);
figure;
plot(x,'k');
hold on;
plot(z,'k');
title('100 draws from N(2,5)');

xbar = mean(x); %2.265487480783972
xmedian = median(x); % 2.310105346957458

s = std(x); %2.183299902847172
s2 = var(x); %4.766798465772472

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%1. b.
%m x n matrix
m = 100;
n = 3;
x = zeros(m,n);

%mean and variance
mu = [0.98 ; -1.44 ; 0.99];
sigma = [1, 0.5, -0.1 ; 0.5, 2, 0.3; -0.1, 0.3, 1.5];
p = chol(sigma)';
testsigma = p*p';

%generate the random draws
for j = 1:1:100
    x(j,1:3) = (mu + p*randn(3,1))';
end

%plot the vector
figure;
plot(x(:,1),'k');
hold on;
plot(x(:,2),'k-o');
plot(x(:,3),'k-+');
plot(z,'k');
title('100 draws from N(\mu,\Sigma)');

%xbar is a 1x3 mean and median of the columns
xbar = mean(x); % [0.7866,-1.4172,1.2386;]
xmedian = median(x); % [0.7794,-1.3346,1.2339;]
s = std(x); % [0.9997,1.4362,1.1181;]

%covariance of the second and third vectors
% note this is a 2x2, 1st and 2nd rows 
% are divided by ";"
cov23 = cov(x(:,2),x(:,3)); % [2.0628,0.2907; 0.2907,1.2502;]
% but we only care about the off-diagonal elements because the variances
% are on the diagonal.

%pearson's correlation coefficient is:
p23 = cov23/(s(1,2)*s(1,3)); %2x2 matrix [1.2845,0.181;0.181,0.7785;]
% and we only car about "0.181"

%alternatively we could use the built in function to calculate pearson's
% but we get the same result.
p23builtin = corr(x(:,2),x(:,3)); % 0.181034440554344

%Here are the correlation coefficients among all the series.
R = corrcoef(x); % 3x3 matrix [1,0.3619,-0.0678;0.3619,1,0.181;-0.0678,0.181,1;]