[MPEI] Aula04

Signed-off-by: Tiago Garcia <tiago.rgarcia@ua.pt>

3ano/1semestre/mpei/aula04/ex1.m

@@ -0,0 +1,53 @@
+% Joint probability distribution
+P_XY = [0.3, 0.2, 0; 
+        0.1, 0.15, 0.05; 
+        0, 0.1, 0.1];
+
+% Values of X and Y
+X_vals = [0, 1, 2];
+Y_vals = [0, 1, 2];
+
+% Marginal distributions
+P_X = sum(P_XY, 2);
+P_Y = sum(P_XY, 1);
+
+% Mean of X and Y
+E_X = sum(X_vals .* P_X');
+E_Y = sum(Y_vals .* P_Y);
+
+% Variance of X and Y
+Var_X = sum((X_vals - E_X).^2 .* P_X');
+Var_Y = sum((Y_vals - E_Y).^2 .* P_Y);
+
+% Covariance of X and Y
+E_XY = sum(sum((X_vals' * Y_vals) .* P_XY));
+Cov_XY = E_XY - E_X * E_Y;
+
+% Correlation coefficient
+rho_XY = Cov_XY / sqrt(Var_X * Var_Y);
+
+% Display results
+fprintf('Marginal PMF of X: \n');
+disp(P_X');
+fprintf('Marginal PMF of Y: \n');
+disp(P_Y);
+fprintf('Mean of X: %.2f\n', E_X);
+fprintf('Mean of Y: %.2f\n', E_Y);
+fprintf('Variance of X: %.2f\n', Var_X);
+fprintf('Variance of Y: %.2f\n', Var_Y);
+fprintf('Covariance of X and Y: %.2f\n', Cov_XY);
+fprintf('Correlation coefficient between X and Y: %.2f\n', rho_XY);
+
+% Plot Marginal PMFs
+figure;
+subplot(1,2,1);
+stem(X_vals, P_X, 'filled');
+title('Marginal PMF of X');
+xlabel('X');
+ylabel('P(X)');
+
+subplot(1,2,2);
+stem(Y_vals, P_Y, 'filled');
+title('Marginal PMF of Y');
+xlabel('Y');
+ylabel('P(Y)');

3ano/1semestre/mpei/aula04/ex2.m

@@ -0,0 +1,42 @@
+% Joint probability distribution
+P_XY = [1/8, 1/8, 1/24; 
+        1/8, 1/4, 1/8; 
+        1/24, 1/8, 1/24];
+
+% Values of X and Y
+X_vals = [-1, 0, 1];
+Y_vals = [-1, 0, 1];
+
+% Marginal distributions
+P_X = sum(P_XY, 2); % Sum across rows for X
+P_Y = sum(P_XY, 1); % Sum across columns for Y
+
+% Checking independence
+independence = true; % Assume independence initially
+
+for i = 1:length(X_vals)
+    for j = 1:length(Y_vals)
+        % Calculate product of marginals
+        product_marginals = P_X(i) * P_Y(j);
+        
+        % Compare with joint probability
+        if abs(P_XY(i, j) - product_marginals) > 1e-10
+            independence = false;
+            break;
+        end
+    end
+    if ~independence
+        break;
+    end
+end
+
+% Display results
+fprintf('Marginal PMF of X: \n');
+disp(P_X');
+fprintf('Marginal PMF of Y: \n');
+disp(P_Y);
+if independence
+    fprintf('X and Y are independent.\n');
+else
+    fprintf('X and Y are not independent.\n');
+end

3ano/1semestre/mpei/aula04/ex3.m

@@ -0,0 +1,72 @@
+%% a)
+% Number of students
+num_students = 120;
+
+% Mean and variance for N1 and N2
+mu_N1 = 14;
+var_N1 = 3.5;
+mu_N2 = 16.8;
+var_N2 = 4.2;
+
+% Standard deviations
+sigma_N1 = sqrt(var_N1);
+sigma_N2 = sqrt(var_N2);
+
+% Generate N1 and N2 from a normal distribution
+N1 = round(normrnd(mu_N1, sigma_N1, [num_students, 1]));
+N2 = round(normrnd(mu_N2, sigma_N2, [num_students, 1]));
+
+% Ensure values are within a reasonable range (e.g., scores between 0 and 20)
+N1 = max(0, min(20, N1));
+N2 = max(0, min(20, N2));
+
+%% b)
+% Joint PMF calculation
+joint_counts = zeros(21, 21);
+for i = 1:num_students
+    joint_counts(N1(i)+1, N2(i)+1) = joint_counts(N1(i)+1, N2(i)+1) + 1;
+end
+joint_pmf = joint_counts / num_students;
+
+% Plotting the joint PMF
+[X, Y] = meshgrid(0:20, 0:20);
+figure;
+bar3(joint_pmf);
+xlabel('N_1 scores');
+ylabel('N_2 scores');
+zlabel('Joint Probability');
+title('Joint PMF of N_1 and N_2');
+
+%% c)
+% Calculate correlation coefficient
+correlation_matrix = corrcoef(N1, N2);
+correlation_coefficient = correlation_matrix(1, 2);
+
+% Display result
+fprintf('Correlation coefficient between N1 and N2: %.2f\n', correlation_coefficient);
+
+%% d)
+% Marginal PMFs
+marginal_N1 = sum(joint_pmf, 2);
+marginal_N2 = sum(joint_pmf, 1);
+
+% Check independence by verifying if joint PMF = product of marginals
+independent = true;
+for i = 1:21
+    for j = 1:21
+        if abs(joint_pmf(i, j) - (marginal_N1(i) * marginal_N2(j))) > 1e-10
+            independent = false;
+            break;
+        end
+    end
+    if ~independent
+        break;
+    end
+end
+
+% Display independence result
+if independent
+    fprintf('N1 and N2 are independent.\n');
+else
+    fprintf('N1 and N2 are not independent.\n');
+end

3ano/1semestre/mpei/aula04/ex4.m

@@ -0,0 +1,22 @@
+% Given probabilities
+P_S = 0.75;       % Probability of a sunny day
+P_R = 0.25;       % Probability of a rainy day
+P_C_given_S = 1/3;  % Probability meteorologist is correct given sunny day
+P_C_given_R = 1;    % Probability meteorologist is correct given rainy day
+
+% Part 1: Calculate overall accuracy of the meteorologist
+P_C = P_C_given_S * P_S + P_C_given_R * P_R;
+
+% Part 2: Calculate accuracy if the student always predicts sun
+P_student_correct = P_S;
+
+% Display the results
+fprintf('Meteorologist''s overall accuracy: %.2f%%\n', P_C * 100);
+fprintf('Student''s accuracy if they always predict sun: %.2f%%\n', P_student_correct * 100);
+
+% Decision analysis
+if P_C > P_student_correct
+    fprintf('The meteorologist''s balanced prediction method is better.\n');
+else
+    fprintf('The student''s method of always predicting sun is better.\n');
+end