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
- A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function
$ f(x) = \begin{cases} e^{-x}, & \text{if } x>0, \\ 0, & \text{otherwise,} \end{cases} $
and that of Brand II follows a gamma distribution with the probability density function
$ g(x) = \begin{cases} \frac{x}{4} e^{-x/2}, & \text{if } x>0, \\ 0, & \text{otherwise.} \end{cases} $
The company uses the batteries of Brands I and II in proportion of $20\%$ and $80\%$ respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more that $2$ years is - Consider a discrete random variable $X$ with the probability mass function
$ P(X = 0) = \frac{\theta}{3}, \ P(X = 1) = 1 - \frac{\theta}{2}, \ P(X = 2) = \frac{\theta}{6}, $
where $\theta \in (0,1)$ is an unknown parameter. In a random sample of size $90$ from this distribution, the observed counts for $X = 0, 1$ and $2$ are $20, 60$ and $10$, respectively. Then, the maximum likelihood estimate of $\theta$ is - Let $X$ be a random sample of size $1$ from the probability density function
$ f(x|\theta) = \begin{cases} \frac{3}{\theta^3} (\theta - x)^2, & \text{if } 0<x<\theta, \\ 0, & \text{otherwise.} \end{cases} $
If $ \left(\frac{X}{1-\lambda_1}, \frac{X}{1-\lambda_2}\right) $ is a confidence interval for $\theta$ with confidence coefficient $1 - \alpha$, where $\lambda_i \in (0,1), \ i = 1,2, \ \lambda_1<\lambda_2$, and $\alpha \in (0,1)$, then which of the following statements is true? - Let $X_1, X_2, . . ., X_n$ be a random sample from a continuous distribution with the common probability density function
$ f(x|\theta) = \begin{cases} \frac{2\theta^2}{x^{\theta+1}}, & \text{if } x>2, \\ 0, & \text{otherwise,} \end{cases} $
where $\theta (> 0)$ is an unknown parameter. Suppose $P(Y>\chi^2_{m,\beta}) = \beta$, where $Y \sim \chi^2_m$. For testing $H_0: \theta = 1$ against $H_1 : \theta>1$, a uniformly most powerful test of size $\alpha, \ 0<\alpha<1$, will reject $H_0$ if - Suppose we want to estimate the population mean $\bar{Y}$ of a variable for a finite population of size $85$, with $34$ Statisticians and $51$ Biologists. We consider the following sampling scheme:
A stratified random sample with $2$ strata of Statisticians (Stratum-1) and Biologists (Stratum-2), where $12$ Statisticians and $15$ Biologists are drawn from Stratum-1 and Stratum-2, respectively, using SRSWOR scheme.
Denote $\bar{y}_S, \bar{y}_B$, and $\bar{y}$ as the mean of the variable among the Statistician sample, Biologist sample, and the combined sample, respectively. Which of the following is an unbiased estimator of $\bar{Y}$?
