1. Let $$ \left\{X_{n} ight\} $$ be a sequence of independent identically distributed random variables following $$ R(0, heta) $$. Show whether $$ X_{n}=\max _{i=1,2, \ldots, n}\left\{X_{i} ight\} $$ convergences in law. 2. Show the order statistic $$ X_{(r)}, X_{(S)}, r

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1. Let $$ \left\{X_{n}ight\} $$ be a sequence of independent identically distributed random variables following $$ R(0, 	heta) $$. Show whether $$ X_{n}=\max _{i=1,2, \ldots, n}\left\{X_{i}ight\} $$ convergences in law. 2. Show the order statistic $$ X_{(r)}, X_{(S)}, r

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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
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 $$ X_1 $$.

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