请你运用matlab代码，解决里面的问题

好的，我可以帮助你编写MATLAB代码来解决《第二次上机作业.pdf》中的问题。以下是每个问题的详细代码实现：

### 问题1：生成随机变量X和Y

% 生成1000个服从均匀分布U(0,3)的随机变量X
X = rand(1, 1000) * 3;

% 利用Y = -2X + 1生成1000个Y
Y = -2 * X + 1;

### 问题2：计算X和Y的方差、协方差和相关系数

% 计算X和Y的方差
DX = var(X);
DY = var(Y);

% 计算X和Y的协方差
CovXY = cov(X, Y);

% 计算X和Y的相关系数
CorrXY = corrcoef(X, Y);

% 理论值
DX_theory = (3^2) / 12; % 均匀分布U(0,3)的方差
DY_theory = (-2)^2 * DX_theory; % Y = -2X + 1的方差
CovXY_theory = -2 * DX_theory; % 协方差
CorrXY_theory = -2 * sqrt(DX_theory) / sqrt(DY_theory); % 相关系数

% 输出结果
fprintf('D(X) = %.4f, D(X)_theory = %.4f\n', DX, DX_theory);
fprintf('D(Y) = %.4f, D(Y)_theory = %.4f\n', DY, DY_theory);
fprintf('Cov(X, Y) = %.4f, Cov(X, Y)_theory = %.4f\n', CovXY(1,2), CovXY_theory);
fprintf('Corr(X, Y) = %.4f, Corr(X, Y)_theory = %.4f\n', CorrXY(1,2), CorrXY_theory);

### 问题3：讨论不同n值下的协方差和相关系数变化

ns = [0, 1, 5, 20, 50];
for n = ns
    Y_n = 2 * X.^n + 1;
    CovXY_n = cov(X, Y_n);
    CorrXY_n = corrcoef(X, Y_n);
    
    fprintf('n = %d: Cov(X, Y) = %.4f, Corr(X, Y) = %.4f\n', n, CovXY_n(1,2), CorrXY_n(1,2));
end

### 问题4：验证统计量X1^2 + X2^2的分布形式

% 生成1000个标准正态随机变量
X1 = normrnd(0, 1, 1, 1000);
X2 = normrnd(0, 1, 1, 1000);

% 计算统计量X1^2 + X2^2
T = X1.^2 + X2.^2;

% 画出密度函数曲线
figure;
histogram(T, 'Normalization', 'pdf');
hold on;
x = linspace(min(T), max(T), 1000);
plot(x, chi2pdf(x, 2), 'r-', 'LineWidth', 2);
title('Density Function of T = X1^2 + X2^2');
legend('Empirical PDF', 'Theoretical Chi-Square PDF');

% 验证矩关系
mean_T = mean(T);
var_T = var(T);

fprintf('Mean of T = %.4f, Theoretical Mean = %.4f\n', mean_T, 2);
fprintf('Variance of T = %.4f, Theoretical Variance = %.4f\n', var_T, 4);

### 问题5：生成标准正态随机变量和泊松随机向量

% (a) 生成1000个标准正态随机变量
Z = normrnd(0, 1, 1, 1000);

% 计算样本期望和样本方差
mu_Z = mean(Z);
sigma2_Z = var(Z);

% 画直方图
figure;
histogram(Z, 'Normalization', 'pdf');
hold on;
x = linspace(min(Z), max(Z), 1000);
plot(x, normpdf(x, 0, 1), 'r-', 'LineWidth', 2);
title('Histogram of Standard Normal Random Variables');
legend('Empirical PDF', 'Theoretical Normal PDF');

% (b) 生成泊松随机向量
lambda = 2;
XX = poissrnd(lambda, 1, 100);

% 计算样本均值和样本方差
mu_XX = mean(XX);
sigma2_XX = var(XX);

% 画直方图
figure;
histogram(XX, 'Normalization', 'pdf');
hold on;
x = 0:10;
plot(x, poisspdf(x, lambda), 'r-', 'LineWidth', 2);
title('Histogram of Poisson Random Vector');
legend('Empirical PDF', 'Theoretical Poisson PDF');

% (c) 重复生成泊松随机向量1000次
num_samples = 1000;
Y = zeros(num_samples, 1);
for i = 1:num_samples
    XX_i = poissrnd(lambda, 1, 100);
    Y(i) = mean(XX_i) - 2;
end

% 标准化Y
SY = Y ./ sqrt(2);

% 计算SY的均值和方差
mu_SY = mean(SY);
sigma2_SY = var(SY);

% (d) 画SY的直方图
figure;
histogram(SY, 'Normalization', 'pdf');
hold on;
x = linspace(min(SY), max(SY), 1000);
plot(x, normpdf(x, 0, 1), 'r-', 'LineWidth', 2);
title('Histogram of Standardized Y');
legend('Empirical PDF', 'Theoretical Normal PDF');

% 比较与标准正态分布
figure;
subplot(1, 2, 1);
histogram(Z, 'Normalization', 'pdf');
hold on;
plot(x, normpdf(x, 0, 1), 'r-', 'LineWidth', 2);
title('Standard Normal Distribution');
legend('Empirical PDF', 'Theoretical Normal PDF');

subplot(1, 2, 2);
histogram(SY, 'Normalization', 'pdf');
hold on;
plot(x, normpdf(x, 0, 1), 'r-', 'LineWidth', 2);
title('Standardized Y');
legend('Empirical PDF', 'Theoretical Normal PDF');

### 问题6：计算最大似然估计值

% 使用5(a)中的样本Z
mu_hat = mean(Z);
sigma2_hat = var(Z);

% 真实参数值
mu_true = 0;
sigma2_true = 1;

% 计算距离
distance_mu = abs(mu_hat - mu_true);
distance_sigma2 = abs(sigma2_hat - sigma2_true);

fprintf('MLE estimate for μ = %.4f, Distance to true μ = %.4f\n', mu_hat, distance_mu);
fprintf('MLE estimate for σ^2 = %.4f, Distance to true σ^2 = %.4f\n', sigma2_hat, distance_sigma2);

以上代码涵盖了所有问题的要求，你可以运行这些代码来完成作业。如果有任何问题或需要进一步的帮助，请随时告诉我！