Region \( S_{\theta} \) Definition
The square region \( S_{\theta} \) has vertices \( (\theta, 0), (0, \theta), (-\theta, 0), (0, -\theta) \). This region is described by the inequality \( |x| + |y| \leq \theta \). The area of this region is \( A = 2\theta^2 \). The joint probability density function (PDF) for \( (X, Y) \) uniformly distributed on \( S_{\theta} \) is: $ f(x, y) = \frac{1}{2\theta^2} \quad \text{for } |x| + |y| \leq \theta $ and \( f(x, y) = 0 \) otherwise.
Statement A Analysis: Uncorrelated Variables
Uncorrelatedness requires \( \text{Cov}(X, Y) = E[XY] - E[X]E[Y] = 0 \). The region \( S_{\theta} \) is symmetric about the origin and axes, making \( E[X] = 0 \) and \( E[Y] = 0 \). $ E[X] = \iint_{S_{\theta}} x f(x, y) \,dx\,dy = 0 $ $ E[Y] = \iint_{S_{\theta}} y f(x, y) \,dx\,dy = 0 $ The expectation \( E[XY] \) is calculated as: $ E[XY] = \iint_{S_{\theta}} xy \left(\frac{1}{2\theta^2}\right) \,dx\,dy $ Since \( xy \) is an odd function in both \( x \) and \( y \), and the integration region \( S_{\theta} \) is symmetric about both axes, the integral evaluates to zero: $ \iint_{S_{\theta}} xy \,dx\,dy = 0 $ Thus, \( E[XY] = 0 \). As \( E[X] = 0 \) and \( E[Y] = 0 \), \( E[XY] = E[X]E[Y] \). \( X \) and \( Y \) are uncorrelated. Statement A is true.
Statement B Analysis: Independent Variables
Independence requires \( f(x, y) = f_X(x) f_Y(y) \). The marginal PDF for \( X \) is derived by integrating \( f(x, y) \) over \( y \): $ f_X(x) = \int_{-(\theta - |x|)}^{\theta - |x|} \frac{1}{2\theta^2} \,dy = \frac{2(\theta - |x|)}{2\theta^2} = \frac{\theta - |x|}{\theta^2} \quad \text{for } |x| \leq \theta $ Similarly, \( f_Y(y) = \frac{\theta - |y|}{\theta^2} \) for \( |y| \leq \theta \). The product of marginal PDFs is: $ f_X(x) f_Y(y) = \frac{(\theta - |x|)(\theta - |y|)}{\theta^4} $ This product does not equal the joint PDF \( f(x, y) = \frac{1}{2\theta^2} \) for all \( (x, y) \in S_{\theta} \). For example, at \( (x, y) = (0, 0) \), \( f_X(0)f_Y(0) = \frac{1}{\theta^2} \neq f(0, 0) \). Therefore, \( X \) and \( Y \) are not independent. Statement B is false.
Statement C Analysis: Bound on \( |X| + |Y| \)
The region \( S_{\theta} \) is defined as \( \{(x, y) | |x| + |y| \leq \theta \} \). Since \( (X, Y) \) is drawn from a uniform distribution over \( S_{\theta} \), any realization \( (x, y) \) must satisfy \( |x| + |y| \leq \theta \). Hence, \( |X| + |Y| \) cannot exceed \( \theta \). Statement C is true.
Statement D Analysis: Maximum Likelihood Estimate (MLE)
Consider two independent observations \( (x_1, y_1) \) and \( (x_2, y_2) \). The likelihood function \( L(\theta) \) is: $ L(\theta) = f(x_1, y_1; \theta) \cdot f(x_2, y_2; \theta) $ $ L(\theta) = \left( \frac{1}{2\theta^2} \mathbf{1}_{|x_1| + |y_1| \leq \theta} \right) \left( \frac{1}{2\theta^2} \mathbf{1}_{|x_2| + |y_2| \leq \theta} \right) $ $ L(\theta) = \frac{1}{4\theta^4} \mathbf{1}_{\max(|x_1| + |y_1|, |x_2| + |y_2|) \leq \theta}} $ To maximize \( L(\theta) \), we need \( \theta \) such that \( \theta \geq \max(|x_1| + |y_1|, |x_2| + |y_2|) \). Since \( \frac{1}{4\theta^4} \) is maximized when \( \theta \) is minimized, the maximum likelihood estimate occurs at the smallest possible \( \theta \). Given \( \theta > 0 \), the minimum value is \( \theta_{MLE} = \max(|x_1| + |y_1|, |x_2| + |y_2|) \). The MLE statement is correct. Statement D is true.
Important Questions from Random Variables
Let $X$, $Y$, and $Z$ be independent Normal random variables with means $-1$, $0$, and $1$, respectively, and variances $1$, $1$, and $3$, respectively. Which of the following random variables has a Cauchy distribution with location parameter $0$ and scale parameter $1$?
- Consider a finite population of size $N = 100$. Let $T_1$ be the sample mean of a study variable based on a sample of size $n$ ($1 < n < N$) under simple random sampling with replacement scheme. Let $T_2$ be the sample mean of the same study variable based on a sample of size $n$ under simple random sampling without replacement scheme. If $Var(T_1) = 9Var(T_2)$, then the sample size $n$ equals
Let $X_1$ and $X_2$ be a random sample from Uniform$[0, \theta]$ distribution, where $\theta > 0$. For testing the hypothesis
$H_0: \theta = 1$ against $H_1: \theta = 2$,
consider a test which rejects $H_0$ if $X_1 + X_2 > \frac{4}{5}$. Then, the probability of type-I error is- Let $\{Y_n: n \ge 1\}$ be a sequence of independent and identically distributed random variables, where $Y_1 \sim \text{Bernoulli}(\frac{1}{2})$. Define $Z = \sum_{n=1}^\infty \frac{4Y_n}{5^n}$. Then, which of the following statements is true?
Let $X$ be a single sample from an absolutely continuous distribution with probability density function
$f(x|\theta) = \begin{cases} \frac{2}{\theta^2}(\theta - x), & \text{if } 0 < x < \theta \\ 0, & \text{otherwise,} \end{cases}$
where $\theta > 0$ is unknown. Which of the following intervals is a $95\%$ confidence interval for $\theta$?
