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zKfM6ferJfl&y~1q0>202Lr+tQc;L+6Om(3sD2%@lJ4z5C%;%?*KPUhD9Q-av3O+cP zNC0$9@c~rg&vYpwnebEgaLhlv`k(N6m*Ygz-!rWMJ$^;ZUlz^85q*LPL_0hskO~+X z@i!$XAS{qf1^6{H?I=WopHC=%hu{dlD?rB?a0B1~l7D*uIu3qOL;$W2=vd$YHzR^B z3aO7m>Jhz(7^Dx%n}mQNF&JGm|96Abhhqp>BLqPY4fpo(K@tfh0t`=r>-wO*VfuP{ z7y=SQ#OPu@{we#*#~UPYz|{12QK19CU3r|?a)NpO5^(%22$#YY%;?=b0`5_OUE793 z= 2) +P_le_1 = pX_teorico(1) + pX_teorico(2); % P(X <= 1) +P_between_1_and_3 = pX_teorico(2) + pX_teorico(3) + pX_teorico(4); % P(1 <= X <= 3) + +% Exibindo os resultados +fprintf('Probabilidades:\n'); +fprintf('P(X >= 2) = %.4f\n', P_ge_2); +fprintf('P(X <= 1) = %.4f\n', P_le_1); +fprintf('P(1 <= X <= 3) = %.4f\n', P_between_1_and_3); diff --git a/3ano/1semestre/mpei/aula03/ex4.m b/3ano/1semestre/mpei/aula03/ex4.m new file mode 100644 index 0000000..162f457 --- /dev/null +++ b/3ano/1semestre/mpei/aula03/ex4.m @@ -0,0 +1,48 @@ +% Parâmetros do problema +n = 5; % número de peças +p = 0.3; % probabilidade de uma peça ser defeituosa +num_simulacoes = 10000; % número total de simulações + +% Simulação +defeituosas = binornd(n, p, num_simulacoes, 1); + +% Cálculo da PMF +pmf = histcounts(defeituosas, 0:n+1, 'Normalization', 'probability'); + +% Cálculo da CDF +cdf = cumsum(pmf); + +% Cálculo da probabilidade de no máximo 2 defeituosas +prob_max_2 = sum(pmf(1:3)); % PMF para 0, 1 e 2 peças defeituosas + +% Exibição dos resultados +disp('PMF:'); +disp(pmf); +disp('CDF:'); +disp(cdf); +disp(['Probabilidade de no máximo 2 peças defeituosas: ', num2str(prob_max_2)]); + +% Gráfico na mesma figura, gráficos diferentes +figure; + +% Subplot para a PMF +subplot(2, 1, 1); % 2 linhas, 1 coluna, 1º gráfico +bar(0:n, pmf, 'FaceColor', 'b', 'EdgeColor', 'k'); +title('Função Massa de Probabilidade (PMF)'); +xlabel('Número de Peças Defeituosas'); +ylabel('Probabilidade'); +xlim([-0.5 n + 0.5]); +grid on; + +% Subplot para a CDF +subplot(2, 1, 2); % 2 linhas, 1 coluna, 2º gráfico +cdf_plot = [0; cdf(:)]; % Adiciona um zero no início +x_values = 0:n; % Valores do eixo x +x_values = [0; x_values(:)]; % Adiciona um zero no início + +plot(x_values, cdf_plot, '-o', 'Color', 'r'); +title('Função Distribuição Acumulada (CDF)'); +xlabel('Número de Peças Defeituosas'); +ylabel('Probabilidade Acumulada'); +xlim([-0.5 n + 0.5]); +grid on; diff --git a/3ano/1semestre/mpei/aula03/ex5.m b/3ano/1semestre/mpei/aula03/ex5.m new file mode 100644 index 0000000..73ca53c --- /dev/null +++ b/3ano/1semestre/mpei/aula03/ex5.m @@ -0,0 +1,22 @@ +% Valores de p variando de 0.001 a 0.5 +p = logspace(-3, log10(0.5), 100); + +% Probabilidades de despenho +P_2_motores = p.^2; +P_4_motores = 4 * p.^3 .* (1 - p) + p.^4; + +% Plotando os resultados +figure; +plot(p, P_2_motores, 'b-', 'LineWidth', 2); +hold on; +plot(p, P_4_motores, 'r-', 'LineWidth', 2); +hold off; + +% Configurações do gráfico +xlabel('Probabilidade de falha (p)'); +ylabel('Probabilidade de despenho'); +title('Probabilidade de Despenho vs. Probabilidade de Falha'); +legend('2 Motores', '4 Motores'); +grid on; +set(gca, 'XScale', 'log'); +ylim([0 1]); diff --git a/3ano/1semestre/mpei/aula03/ex6.m b/3ano/1semestre/mpei/aula03/ex6.m new file mode 100644 index 0000000..9b33c05 --- /dev/null +++ b/3ano/1semestre/mpei/aula03/ex6.m @@ -0,0 +1,21 @@ +% Parâmetros +n = 8000; +p = 1/1000; +k = 7; + +% (a) Cálculo usando a distribuição binomial +% Cálculo da combinação diretamente +comb = 1; +for i = 1:k + comb = comb * (n - (i - 1)) / i; % Cálculo da combinação n!/(k!(n-k)!) +end + +P_binomial = comb * (p^k) * ((1 - p)^(n - k)); + +% (b) Cálculo usando a aproximação de Poisson +lambda = 8; % média de defeituosos em 8000 chips +P_poisson = (lambda^k * exp(-lambda)) / factorial(k); + +% Resultados +fprintf('Probabilidade (Binomial) de 7 defeituosos: %.6f\n', P_binomial); +fprintf('Probabilidade (Poisson) de 7 defeituosos: %.6f\n', P_poisson); diff --git a/3ano/1semestre/mpei/aula03/ex7.m b/3ano/1semestre/mpei/aula03/ex7.m new file mode 100644 index 0000000..9bb73f0 --- /dev/null +++ b/3ano/1semestre/mpei/aula03/ex7.m @@ -0,0 +1,14 @@ +% Parâmetros +lambda = 60; % média em 4 segundos + +% (a) Probabilidade de não receber nenhuma mensagem +P_0 = exp(-lambda); % P(X = 0) + +% (b) Probabilidade de receber mais de 40 mensagens +% Usando a soma das probabilidades de 0 a 40 +P_leq_40 = sum(exp(-lambda) * (lambda.^(0:40)) ./ factorial(0:40)); % P(X <= 40) +P_gt_40 = 1 - P_leq_40; % P(X > 40) + +% Resultados +fprintf('Probabilidade de não receber nenhuma mensagem: %.10f\n', P_0); +fprintf('Probabilidade de receber mais de 40 mensagens: %.10f\n', P_gt_40); diff --git a/3ano/1semestre/mpei/aula03/ex8.m b/3ano/1semestre/mpei/aula03/ex8.m new file mode 100644 index 0000000..4a7c55d --- /dev/null +++ b/3ano/1semestre/mpei/aula03/ex8.m @@ -0,0 +1,12 @@ +% Parâmetros +lambda = 2; % média de erros em 100 páginas + +% Cálculo das probabilidades +P_0 = exp(-lambda); % P(X = 0) +P_1 = (lambda^1 * exp(-lambda)) / factorial(1); % P(X = 1) + +% Probabilidade de no máximo 1 erro +P_leq_1 = P_0 + P_1; + +% Resultados +fprintf('Probabilidade de no máximo 1 erro em 100 páginas: %.6f\n', P_leq_1); diff --git a/3ano/1semestre/mpei/aula03/ex9.m b/3ano/1semestre/mpei/aula03/ex9.m new file mode 100644 index 0000000..7fe0829 --- /dev/null +++ b/3ano/1semestre/mpei/aula03/ex9.m @@ -0,0 +1,31 @@ +% Parâmetros +mu = 14; % média +sigma = 2; % desvio padrão +n_samples = 100000; % número de amostras + +% Gerar amostras da distribuição normal +X = mu + sigma * randn(n_samples, 1); % amostras + +% (a) Probabilidade de um aluno ter classificação entre 12 e 16 +P_a = mean(X >= 12 & X <= 16); + +% (b) Probabilidade de os alunos terem classificações entre 10 e 18 +P_b = mean(X >= 10 & X <= 18); + +% (c) Probabilidade de um aluno passar (classificação maior ou igual a 10) +P_c = mean(X >= 10); + +% Exibir resultados +fprintf('Estimativa da probabilidade (a) entre 12 e 16: %.4f\n', P_a); +fprintf('Estimativa da probabilidade (b) entre 10 e 18: %.4f\n', P_b); +fprintf('Estimativa da probabilidade (c) para passar (>= 10): %.4f\n', P_c); + +% Verificação usando normcdf() +P_a_exact = normcdf(16, mu, sigma) - normcdf(12, mu, sigma); +P_b_exact = normcdf(18, mu, sigma) - normcdf(10, mu, sigma); +P_c_exact = 1 - normcdf(10, mu, sigma); % P(X >= 10) + +% Exibir resultados da verificação +fprintf('Probabilidade exata (a) entre 12 e 16: %.4f\n', P_a_exact); +fprintf('Probabilidade exata (b) entre 10 e 18: %.4f\n', P_b_exact); +fprintf('Probabilidade exata (c) para passar (>= 10): %.4f\n', P_c_exact);