CSIR NET Maths previous year question papers

Let X be a random variable with the cumulative distribution function F_X(x) = 0 if x < 0, F_X(x) = (x + 2)/5 if 0 ≤ x < 2, and F_X(x) = 1 if x ≥ 2. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed random variables with the probability density function f(x|$\theta) =$ $e^{-(x-\theta)}$ if $x \ge \theta$, and 0 otherwise, where $\theta \in \mathbb{R}$ is the unknown parameter. Further, let $Y = \min(X_{1}, X_{2}, \dots, X_{n})$. Which of the following are confidence intervals for $\theta$ with the confidence coefficient $(1 - \alpha)$, where $\alpha \in (0, 1)$?

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{10}, Y_{10})$ be a random sample from a bivariate normal distribution BVN$(\mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho)$ with $\mu_{1} = 5, \mu_{2} = 6, \sigma_{1}^{2} = 4, \sigma_{2}^{2} = 9$ and $\rho = 1/2$. Then, which of the following statements are true?

Let the random vector $(X_{1}, X_{2}, X_{3})^{T}$ have the positive definite dispersion matrix with rows $(1, \rho, \rho), (\rho, 1, \rho)$ and $(\rho, \rho, 1)$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n} (n\ge2)$ be a random sample from Uniform$[-3\theta/2, \theta/2]$ distribution, where $\theta>0$ is an unknown parameter. Let ${\bar{X}}, X_{(1)}$ and $X_{(n)}$ denote respectively the mean, the smallest order statistic and the largest order statistic of the sample. Then which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{10}$ be a random sample of size 10 from a Bernoulli distribution with success probability $1/(1 + e^\theta)$, where $\theta \in \mathbb{R}$ is an unknown parameter. If $M = \sum(i=1..10)X_{i}$, then which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample of size n from $U(\theta, \theta + 1)$ distribution, where $\theta \in \mathbb{R}$ is an unknown parameter. If $T_{n} = \max${$X_{1}, X_{2}, \dots, X_{n}$}, n = 1, 2, …, then which of the following statements are true?

Consider a linear model $Y_{i} = \beta_{1} + \beta_{2} + \cdots + \beta_{i} + \varepsilon_{i}, 1 \le i \le n$, where the errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Then, which of the following statements are true?

Consider maximizing the objective function $P = x_{1} + x_{2}$ subject to $x_{1} + x_{2} + 2x_{3} \le 5, 2x_{1} - 3x_{3} \ge 1, x_{2} + x_{3} \le 0, x_{1} \ge 0, x_{2} \ge 0, x_{3} \ge 0$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample of size n from $U(\theta, \theta + 1)$ distribution, where $\theta \in \mathbb{R}$ is the unknown parameter. If $T_{n} = \min${$X_{1}, X_{2}, \dots, X_{n}$}, n = 1, 2, …, then which of the following statements are true?

Consider a linear model $Y_{i} = \beta_{1} + \beta_{2} + \cdots + \beta_{i} + \varepsilon_{i}, 1 \le i \le n$, where errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Let $\beta$̂$_{i}$ be the best linear unbiased estimator (BLUE) of $\beta_{i}, i = 1, 2, \dots, n$. Then, which of the following statements are true?

Consider a linear regression model $Y = X\beta + \varepsilon$, with r regressors and an intercept. Random error $\varepsilon$ ~ $N_{n}(0, \sigma^{2}I_{n})$ and X has full column rank. Here $I_{n}$ denotes the identity matrix of order n. Regression coefficients are estimated by the least squares estimation method. Let $\sigma$̂$^{2}$ and $\sigma$̂$^{2}($MLE), respectively, be the mean squares residuals and the maximum likelihood estimator of $\sigma^{2}$. Then, which of the following statements are true?

Let $X_{1}, \dots, X_{n} (n \ge 2)$ be a random sample from $a U(-\theta, 2\theta)$ distribution, where $\theta > 0$ is an unknown parameter. Let ${\bar{X}} = (1/n)\sum_{i}$₌$_{1}^{n}X_{i}, X_{(1)} = \min${$X_{1}, \dots, X_{n}$} and $X_{(n)} = \max${$X_{1}, \dots, X_{n}$}. Which of the following statements are true?

Let $Y_{1}, \dots, Y_{n} (n \ge 2)$ be independent observations; $Y_{i}$ ~ $N(\beta{}x_{i}, \sigma^{2}), i = 1, \dots, n$; where $x_{1}, \dots, x_{n}$ and $\sigma^{2}(> 0)$ are known constants and $\beta \in \mathbb{R}$ is an unknown parameter. Consider $N(\beta_{0}, \tau^{2})$ prior for the parameter $\beta$, where $\beta_{0}$ and $\tau^{2}(> 0)$ are known constants, and $N(\mu, \lambda^{2})$ denotes a normal distribution with mean $\mu$ and variance $\lambda^{2}$. Suppose ȳ $= (1/n)\sum_{i}$₌$_{1}^{n}y_{i}$ and ${\bar{x}} = (1/n)\sum_{i}$₌$_{1}^{n}x_{i}$ are observed sample means. Under squared error loss function, which of the following statements are true?

Suppose $X_{1}, X_{2}, \dots$ are independent and identically distributed N(0,1) random variables and $Y_{n} = X_{1}^{4} + X_{2}^{4} + \cdots + X_{n}^{4}$. Which of the following probabilities converge to 1/2 as $n \to \infty$?

Suppose X is a continuous random variable with density $f(x) = (1/\pi)\cdot1/(1 + (x + 1)^{2})$, and Y = X/|X| for X ≠ 0, Y = 0 for X = 0. Then which of the following statements are true?

Let $X_{1}, \dots, X_{n}$ be a random sample with density f(x|$\theta) =$ $e^{\theta-x}$ for $x \ge \theta$, where $\theta \in \mathbb{R}$ is unknown. Define X̄ and $X_{(1)} = \min X_{i}$. Which of the following statements are true?

For n ≥ 2, let $X_{1}, \dots, X_{n}$ be from $N(\mu, \sigma^{2})$ with both unknown, X̄ the mean and $S^{2}$ the sample variance. With $z_\alpha$ and $t_{m,\alpha}$ the $(1 - \alpha)-$th quantiles, which of the following represent 90% confidence intervals for $\mu$?