chi_sq_and_t_distributions.m

%
% T distribution
% --------------------------------------------------------------------------------------------------------------------------

% The t-distribution is made from the means of many random samples. This
% gives you the distribution of where all samples means are likely to fall,
% given the DF. A t-statistic is essentially a Z-score for the distribution,
% t= (Xbar - u) / SE. remember: distributions are generated by random
% numbers. You then set your alpha based on how often you are willing to
% accept a difference by chance. 

%t-distribution DF = 2 
for i = 1:1000000
    X = randn(3,1);
    output_3(i,1) = mean(X);
end

%t-distribution DF = 99 
for i = 1:1000000
    X = randn(100,1);
    output_100(i,1) = mean(X);
end

%normal distrubition N = 1million
for i = 1:1000000
    X = randn(1);
    output_norm(i,1) = X;
end

figure(); histogram(output_3); xlim([-3 3]); title('T-distribution, Df = 2');
figure(); histogram(output_100); xlim([-3 3]); title('T-distribution, Df = 99');
figure(); histogram(output_norm); xlim([-3 3]); title('Normal distribution');


% -----------------------------------------------------------------------------------------------------------------------
% Chi-Sq Distribution
% -----------------------------------------------------------------------------------------------------------------------

% In probability theory and statistics, the chi-square distribution (also chi-squared or ?2-distribution) 
% with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables.

for i = 1:10000
    X = randn(3,1);
    Xbar = mean(X);
    sumOfSquares = sum((X - Xbar).^2);
    chiSq_3(i,1) = sumOfSquares;
end
for i = 1:10000
    X = randn(30,1);
    Xbar = mean(X);
    sumOfSquares = sum((X - Xbar).^2);
    chiSq_30(i,1) = sumOfSquares;
end
for i = 1:10000
    X = randn(100,1);
    Xbar = mean(X);
    sumOfSquares = sum((X - Xbar).^2);
    chiSq_100(i,1) = sumOfSquares;
end
for i = 1:10000
    X = randn(500,1);
    Xbar = mean(X);
    sumOfSquares = sum((X - Xbar).^2);
    chiSq_500(i,1) = sumOfSquares;
end


figure(); histogram(chiSq_3); title('Chisq, Df = 3');
figure(); histogram(chiSq_30); title('Chisq, Df = 30');
figure(); histogram(chiSq_100); title('Chisq, Df = 100');
figure(); histogram(chiSq_500); title('Chisq, Df = 500');