From 11be924274a8928d5d67fe107d84c4f9e7f85452 Mon Sep 17 00:00:00 2001 From: Tiago Garcia Date: Sun, 3 Nov 2024 18:27:10 +0000 Subject: [PATCH] [MPEI] Aula04 Signed-off-by: Tiago Garcia --- ...variáveis_aleatórias_multidimensionais.pdf | Bin 0 -> 60507 bytes 3ano/1semestre/mpei/aula04/ex1.m | 53 +++++++++++++ 3ano/1semestre/mpei/aula04/ex2.m | 42 ++++++++++ 3ano/1semestre/mpei/aula04/ex3.m | 72 ++++++++++++++++++ 3ano/1semestre/mpei/aula04/ex4.m | 22 ++++++ 5 files changed, 189 insertions(+) create mode 100644 3ano/1semestre/mpei/aula04/MPEI_2024_2025_PL04_variáveis_aleatórias_multidimensionais.pdf create mode 100644 3ano/1semestre/mpei/aula04/ex1.m create mode 100644 3ano/1semestre/mpei/aula04/ex2.m create mode 100644 3ano/1semestre/mpei/aula04/ex3.m create mode 100644 3ano/1semestre/mpei/aula04/ex4.m diff --git a/3ano/1semestre/mpei/aula04/MPEI_2024_2025_PL04_variáveis_aleatórias_multidimensionais.pdf 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~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 diff --git a/3ano/1semestre/mpei/aula04/ex3.m b/3ano/1semestre/mpei/aula04/ex3.m new file mode 100644 index 0000000..e408166 --- /dev/null +++ b/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 diff --git a/3ano/1semestre/mpei/aula04/ex4.m b/3ano/1semestre/mpei/aula04/ex4.m new file mode 100644 index 0000000..6e3987c --- /dev/null +++ b/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