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1. Let \( \left\{X_{n}\right\} \) be a sequence of independent identically distributed random variables following \( R(0, \theta) \). Show whether \( X_{n}=\max _{i=1,2, \ldots, n}\left\{X_{i}\right\} \) convergences in law. 2. Show the order statistic \( X_{(r)}, X_{(S)}, r
Ask by Howell Schwartz. in Kenya
Mar 09,2025

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1. The sequence \( X_n = \max\{X_1, X_2, \ldots, X_n\} \) converges in law to a Gumbel distribution as \( n \to \infty \). 2. The correlation coefficient between the \( r \)-th and \( s \)-th order statistics \( X_{(r)} \) and \( X_{(s)} \) from a uniform distribution is: \[ \rho = \left[\frac{r(n-s+1)}{s(n-r+1)}\right]^{1/2} \] 3. As \( n \to \infty \), \( \sqrt{2 \chi_n^2} \) converges in distribution to \( N(\sqrt{2n-1}, 1) \). 4. **Weak Law of Large Numbers (WLLN):** - *Statement:* If \( X_1, X_2, \ldots, X_n \) are independent and identically distributed with finite mean \( \mu \) and variance \( \sigma^2 \), then: \[ P\left(\left|\frac{1}{n}\sum_{i=1}^{n} X_i - \mu\right| \geq \epsilon\right) \to 0 \quad \text{as } n \to \infty \] - *Proof:* Apply Chebyshev's inequality to show that the probability of the sample mean deviating from the population mean by more than \( \epsilon \) becomes negligible as \( n \) increases. 5. **Joint Distributions and Conditional Expectation:** - *(i)* The joint marginal distribution of \( Z = 4y_1 - 2y_2 + y_3 - 3y_4 \) is a normal distribution with mean and variance determined by \( \mu \) and \( \Sigma \). - *(ii)* The joint distribution of \( Z_1 = y_1 + y_2 + y_3 + y_4 \) and \( Z_2 = -2y_1 + 3y_2 - 2y_4 \) is also a bivariate normal distribution with means, variances, and covariance derived from \( \mu \) and \( \Sigma \). - *(iii)* The conditional expectation \( E(X_2 \mid X_1) \) can be found using the properties of multivariate normal distributions, resulting in a linear function of \( X_1 \).

Solución

Let's break down the problem into its components and solve each part step by step. ### 1. Convergence in Law of \( X_n = \max_{i=1,2,\ldots,n} \{X_i\} \) Given that \( \{X_n\} \) is a sequence of independent identically distributed random variables following \( R(0, \theta) \), we want to show whether \( X_n \) converges in law. **Step 1: Distribution of \( X_n \)** The cumulative distribution function (CDF) of \( X_n \) can be expressed as: \[ P(X_n \leq x) = P(\max_{i=1,2,\ldots,n} X_i \leq x) = P(X_1 \leq x)^n \] Let \( F(x) = P(X_1 \leq x) \). Then, \[ P(X_n \leq x) = [F(x)]^n \] **Step 2: Behavior as \( n \to \infty \)** As \( n \to \infty \), if \( F(x) < 1 \), then \( [F(x)]^n \to 0 \). If \( F(x) = 1 \), then \( [F(x)]^n \to 1 \). Therefore, we need to find the limit of \( X_n \) as \( n \to \infty \). **Step 3: Convergence in Law** The limit distribution of \( X_n \) will depend on the behavior of \( F(x) \). If \( \theta \) is finite, \( X_n \) will converge to the distribution of the maximum of the distribution, which is a Gumbel distribution. Thus, \( X_n \) converges in law to a Gumbel distribution. ### 2. Correlation Coefficient of Order Statistics Let \( X_{(r)} \) and \( X_{(s)} \) be the \( r \)-th and \( s \)-th order statistics from a sample of size \( n \) drawn from \( U(0,1) \). **Step 1: Covariance Calculation** The covariance between \( X_{(r)} \) and \( X_{(s)} \) can be calculated using the properties of order statistics. The expected values and variances can be derived from the uniform distribution. **Step 2: Correlation Coefficient Formula** The correlation coefficient \( \rho \) is given by: \[ \rho = \frac{Cov(X_{(r)}, X_{(s)})}{\sqrt{Var(X_{(r)}) Var(X_{(s)})}} \] Using the known results for order statistics from a uniform distribution, we can derive: \[ Cov(X_{(r)}, X_{(s)}) = \frac{r(n-s+1)}{n+1} \] and \[ Var(X_{(r)}) = \frac{r(n-r+1)}{(n+1)^2(n+2)} \] \[ Var(X_{(s)}) = \frac{s(n-s+1)}{(n+1)^2(n+2)} \] **Step 3: Final Correlation Coefficient** Substituting these into the correlation coefficient formula gives: \[ \rho = \left[\frac{r(n-s+1)}{s(n-r+1)}\right]^{1/2} \] ### 3. Convergence of \( \sqrt{2 \chi_n^2} \) We want to show that \( \sqrt{2 \chi_n^2} \stackrel{a}{\rightarrow} N(\sqrt{2n-1}, 1) \) as \( n \to \infty \). **Step 1: Chi-Squared Distribution** The chi-squared distribution \( \chi_n^2 \) with \( n \) degrees of freedom can be approximated by a normal distribution as \( n \) becomes large. **Step 2: Central Limit Theorem Application** By the Central Limit Theorem, we have: \[ \frac{\chi_n^2 - n}{\sqrt{2n}} \xrightarrow{d} N(0, 1) \] Thus, \[ \chi_n^2 \approx n + \sqrt{2n}Z \quad \text{for } Z \sim N(0, 1) \] **Step 3: Transformation** Taking the square root: \[ \sqrt{2 \chi_n^2} \approx \sqrt{2(n + \sqrt{2n}Z)} = \sqrt{2n} + Z \] As \( n \to \infty \), this converges in distribution to \( N(\sqrt{2n-1}, 1) \). ### 4. Weak Law of Large Numbers **Statement:** The Weak Law of Large Numbers states that if \( X_1, X_2, \ldots, X_n \) are independent and identically distributed random variables with finite mean \( \mu \) and variance \( \sigma^2 \), then: \[ P\left(\left|\frac{1}{n}\sum_{i=1}^{n} X_i - \mu\right| \geq \epsilon\right) \to 0 \quad \text{as } n \to \infty \] **Proof:** Using Chebyshev's inequality: \[ P\left(\left|\frac{1}{n}\sum_{i=1}^{n} X_i - \mu\right| \geq \epsilon\right) \leq \frac{Var\left(\frac{1}{n}\sum_{i=1}^{n} X_i\right)}{\epsilon^2} = \frac{\sigma^2}{n\epsilon^2} \] As \( n \to \infty \), the right-hand side approaches 0, proving the weak law. ### 5. Joint Distribution and Conditional Expectation Given \( X \sim N_4(\mu, \Sigma) \) with specified \( \mu \) and \( \Sigma \): **(i) Joint Marginal Distribution of \( Z = 4y_1 - 2y_2 + y_3 - 3y_4 \)** To find the distribution of \( Z \), we compute: \[ Z \sim N(\mu_Z, \sigma_Z^2) \] where \( \mu_Z = 4\mu_1 - 2\mu_2 + \mu_3 - 3\mu_4 \) and \( \sigma_Z^2 \) is derived from the covariance matrix \( \Sigma \). **(ii) Joint Distribution of \( Z_1 = y_1 + y_2 + y_3 + y_4 \) and \( Z_2 = -2y_1 + 3y_2 - 2y_4 \)** The joint

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Extra Insights

Understanding the intricacies of statistical theory can be quite the adventure! For your first query about the maximum \( X_n = \max_{i=1,2,\ldots,n} \{X_i\} \) from independent identically distributed random variables following \( R(0, \theta) \), one can expect that as \( n \) grows larger, the maximum value will reflect the tail behavior of the underlying distribution. Specifically, the distribution of \( X_n \) can be used to show convergence in law, often resulting in the fact that the maximum value approaches the upper boundary of the support of \( R(0, \theta) \). Now, about the correlation coefficient between order statistics \( X_{(r)} \) and \( X_{(s)} \) when samples are drawn from a uniform distribution \( U(0,1) \): the beauty here lies in the symmetry and uniformity of the distribution. The formula provides a neat insight into dependencies among order statistics and reveals how their relative positions in a sample lead to predictable patterns of correlation. As you can see, statistics isn't just about the numbers, it's about the beautiful relationships we can uncover!

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