CSIR NET Maths previous year question papers

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 3)$ be a random sample from a population with absolutely continuous cumulative distribution function F(·). The corresponding order statistics are X(1:n) < X(2:n) < ⋯ < X(r:n) < ⋯ < X(n:n). For r = 2, 3, define Y(r,n) = nF(X(r:n)). Suppose that Y(r,n) converges in distribution to a random variable Y_r as $n \to \infty, r = 2, 3$. Then, which of the following statements are true?

Let $X_{1}$ and $X_{2}$ be random variables having absolutely continuous distribution functions. Let $h_{i}(t)$ denote the hazard function of $X_{i}, i = 1, 2$. If $h_{1}(t) \le h_{2}(t)$ for all $t \in \mathbb{R}$, then which of the following statements are true?

Let X be a random variable with probability density function $f(x) = 1/(\pi(1 + x^{2})), x \in \mathbb{R}$. If $Z = 1/2 + (1/\pi)\tan$⁻$^{1}(X)$, then which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{13}$ be independent and identically distributed (i.i.d.) Poisson$(\theta)$ random variables, where $\theta > 0$. Then, which of the following statements are true?

Consider the following design where the columns represent blocks and the letters represent treatments — block 1: A, B, E; block 2: C, D, E; block 3: A, C, F; block 4: B, D, F; block 5: A, D, G; block 6: B, C, G; block 7: E, F, G. Then, which of the following statements are true?

Consider the linear programming problem: max{$x_{1} + x_{2} + x_{3}$} subject to constraints $x_{1} + x_{2} - x_{3} \le 1, x_{1} + x_{3} \le 2, 0 \le x_{1} \le 1/2, x_{2} \ge 0$, and $0 \le x_{3} \le 1$. Which of the following statements are true?

Let X and Y be independent random variables with X ~ N(2, 4) and Y ~ N(−4, 9) where $N(\mu, \sigma^{2})$ denotes a normal distribution with mean $\mu$ and variance $\sigma^{2}$. Given $\Phi(1) = 0.8413, \Phi(2) = 0.9772$ and $\Phi(3) = 0.9987$ where $\Phi(\cdot)$ is the cumulative distribution function of a standard normal random variable. Which of the following statements are true?

Transition probability matrix of a homogeneous Markov chain with states 0, 1, 2, 3 is P = [[1/4, 3/4, 0, 0], [1, 0, 0, 0], [2/3, 0, 1/3, 0], [0, 0, 2/5, 3/5]], where rows and columns are indexed 0, 1, 2, 3. Which of the following statements are true?

Let X and Y be jointly distributed continuous random variables with joint probability density function f(x, y) = x/y if 0 < x < y < 2, and 0 otherwise. Which of the following statements are true?

Let $X_{1}, X_{2}$ denote lifetimes (in years) of 2 components of an electronic system. Let $Y_{1} = X_{1} + X_{2}, Y_{2} = \max${$X_{1}, X_{2}$} and $Y_{3} = \min${$X_{1}, X_{2}$}. Assume that $X_{1}$ and $X_{2}$ are independent, each following exponential distribution with probability density function f(x) = (1/2)e^(−x/2) if x > 0, and 0 otherwise. Which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}$ be a random sample from a continuous distribution having cumulative distribution function F(t), probability density function f(t), and failure rate function r(t) = f(t)/(1 − F(t)), t > 0, where F(0) = 0. If r(t) = 1 for all t > 0, 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?

Let $X_{1}, \dots, X_{12}$ be a random sample from the N(2, 4) distribution and $Y_{1}, \dots, Y_{15}$ be a random sample from the N(−2, 5) distribution, where $N(\mu, \sigma^{2})$ denotes a normal distribution with mean $\mu$ and variance $\sigma^{2}$. Assume that the two random samples are mutually independent. Let ${\bar{X}} = (1/12)\sum_{i}$₌$_{1}^{12}X_{i}, S_{1}^{2} = (1/11)\sum_{i}$₌$_{1}^{12}(X_{i} - {\bar{X}})^{2}$, Ȳ $= (1/15)\sum_{j}$₌$_{1}^{15}Y_{j}, S_{2}^{2} = (1/14)\sum_{j}$₌$_{1}^{15}(Y_{j} -$ Ȳ$)^{2}$. Which of the following statements are true?

In a standard linear regression model, let $R^{2}$ and ${\bar{R}}^{2}$, respectively, denote the coefficient of determination and adjusted coefficient of determination. Which of the following statements are true?

Consider the two-way ANOVA model $Y_{ij} = \mu + \alpha_{i} + \beta_{j} + \varepsilon_{ij}, i = 1, 2; j = 1, 2$, where $\mu$ is the overall mean effect, $\alpha_{i}$ is the effect of the i-th level of factor $A, \beta_{j}$ is the effect of j-th level of factor $B, Y_{ij}$ is the response of the (i, j)-th experimental unit and $\varepsilon_{ij}$ is the corresponding error with $E(\varepsilon_{ij}) = 0$ for i = 1, 2; j = 1, 2. Which of the following are estimable linear parametric functions?

Let $X = (X_{1}, X_{2})^{T}$ be a bivariate random vector with covariance matrix $\sum = [[1, \sqrt{2}], [\sqrt{2}, 2]]$. Which of the following statements are true?

Let {$X_{n}$}$_{n}\ge_{1}$ be a sequence of independent and identically distributed random variables with $E(X_{1}) = 0$ and $Var(X_{1}) = 1$. Which of the following statements are true?

Let $X_{1}, \dots, X_{n}$ be independent and identically distributed $U(0, \theta), \theta > 0$ random variables. Define $X_{(n)} = \max${$X_{1}, \dots, X_{n}$} and $X_{(1)} = \min${$X_{1}, \dots, X_{n}$}. Which of the following statements are true?

Let $X_{1}, \dots, X_{n} (n \ge 3)$ be a random sample from a distribution having probability density function f(x | $\theta) = \theta{}e^(-\theta{}x)$ if x > 0, and 0 otherwise, where $\theta > 0$ is an unknown parameter. Let $T_{n} = (1/n)\sum_{i}$₌$_{1}^{n}X_{i}$. Which of the following statements are true?

Consider a six faced die whose i-th face is marked with i dots, i = 1, 2, …, 6. In a single random throw of the die, let $p_{i}$ denote the probability that the obtained upper face has i dots, i = 1, 2, …, 6. The die is rolled 240 times independently and the following result is obtained — face observed 1, 2, 3, 4, 5, 6 with frequencies 40, 55, 40, 25, 35, 45 respectively. Suppose we want to test $H_{0} : p_{i} = 1/6$ for i = 1, 2, …, 6; against $H_{1} : p_{i} \ne 1/6$ for at least one i. It is given that $\chi^{2}_{5};_{0}._{05} = 11.07, \chi^{2}_{6};_{0}._{05} = 12.59, \chi^{2}_{5};_{0}._{01} = 15.09, \chi^{2}_{6};_{0}._{01} = 16.81$. Based on the asymptotic goodness of fit $\chi^{2}$ test for testing $H_{0}$ against $H_{1}$, which of the following statements are true?

Observations on the shear strength of concrete from 5 randomly selected structures are given below — structure 1, 2, 3, 4, 5 with shear strength 1718.4, 1787.4, 2562.3, 2356.9, 2153.2 respectively. The null hypothesis $H_{0}$ that the median shear strength is 2000 units is tested against the alternative hypothesis $H_{1}$ that the median shear strength is greater than 2000 units at 5% level of significance. Which of the following statements are true?

Consider the following ANOVA table for a randomized block design — Treatments: sum of squares 48, degrees of freedom 4, mean squares 12, F calculated $\beta$; Blocks: sum of squares 72, degrees of freedom 3, mean squares 24, F calculated 12; Error: sum of squares $\alpha$, degrees of freedom m, mean squares $\gamma$; Total: sum of squares 144, degrees of freedom 19. Which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}, X_{4}$ be i.i.d. Bernoulli(1/3) and $X_{(1)} \le \cdots \le X_{(4)}$ the order statistics. Which of the following is true?

Let X be a Poisson random variable with mean $\lambda$. Which of the following parametric functions is not estimable?