CSIR NET Maths previous year question papers

Suppose that we have a data set consisting of 2n + 1 observations for some $n \in \mathbb{N}$. Value of each observation is either x or x + r, where $x \in \mathbb{N}, r \ge 0$. Then, which of the following statements is always true?

A biased six-faced die is tossed once. Suppose that the probability of any prime number showing up is twice that of any non-prime number showing up. Then, the probability that an odd number will show up is

Let {$Y_{n} : n \ge 1$} be a sequence of independent and identically distributed random variables, where $Y_{1}$ ~ Bernoulli(1/2). Define $Z = \sum_{n=1}^\infty$ $4Y_{n}/5^{n}$. Then, which of the following statements is true?

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?

Suppose that the probability density function of the random variable X is $f(x) = 2(\theta - x)/\theta^{2}$ if $0 \le x \le \theta$, and 0 otherwise, where $\theta > 0$ is an unknown parameter. Based on a single observation X, the confidence coefficient of the confidence interval [(2/5)X, (5/2)X] for $\theta$ is

Suppose that X ~ binomial$(9, \theta), 0.7 < \theta < 1$, and Y ~ Poisson$(\lambda), \lambda > 0$. If 3E(Y) = E(X), then which of the following is true?

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 2)$ be a random sample from Uniform$(\theta, 1 - \theta)$ distribution, where $-\infty < \theta < 1/2$. The maximum likelihood estimator of $\theta$ is

Let X be a random variable with cumulative distribution function given by F(x) = 0 if x < 0; F(x) = (x + 1)/3 if 0 ≤ x < 1; F(x) = 1 if x ≥ 1. Then the value of P(1/3 < X < 3/4) + P(X = 0) is equal to

Let $X_{1}, X_{2}$ be a random sample from a population having probability density function f ∈ {$f_{0}, f_{1}$} where $f_{0}(x) = 1/2$ if 0 ≤ x ≤ 2 and 0 otherwise, and $f_{1}(x) = 1/4$ if 0 ≤ x ≤ 4 and 0 otherwise. For testing the null hypothesis $H_{0} : f = f_{0}$ against the alternate hypothesis $H_{1} : f = f_{1}$, the power of a most powerful test of size $\alpha = 0.05$ is equal to

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?