CSIR NET Maths previous year question papers

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 manufacturing unit producing three-component series systems, where component's lifetimes are independent and identically distributed random variables with hazard function $h(x) = \lambda$ if x > 0, and 0 otherwise. Here, $\lambda > 0$ is an unknown parameter. Let $X_{1}, X_{2}, X_{3}$ be a random sample of size 3 on lifetimes of systems produced by the manufacturing unit. Let $\theta = P_\lambda(X_{1} > 1/2)$. If $\theta$̂ is the maximum likelihood estimate of $\theta$ based on the realization $x_{1} = 1/2, x_{2} = 1/6, x_{3} = 1/3$ of the random sample, then which of the following statements are true?

The life (in hours) of an electrical component is exponentially distributed with mean $\theta > 0$. For testing $H_{0} : \theta = 5/(\ln(10) - \ln(9))$ against $H_{1} : \theta = 5/\ln(2)$, three such components are chosen at random. Suppose that we reject the null hypothesis $H_{0}$ if and only if two or more of these three components survive for less than five hours. 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 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?

If $(X_{1}, X_{2})$ᵗ ~ $N_{2}((0, 0)$ᵗ, [[3, 1], [1, 3]]), then which of the following statements are true?

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). Define, for r = 2, 3, 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, Y, Z be independent and identically distributed (i.i.d.) random variables, each following Bernoulli$(\theta), 0 < \theta < 1$. Then, which of the following statements are true?

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{n}, Y_{n})$ be a random sample of size n from a bivariate distribution F(X,Y) with absolutely continuous marginal distribution functions F_X and F_Y of X and Y, respectively. Let r_S be the Spearman's rank correlation coefficient defined with ranks of $X_{i}$ and ranks of $Y_{i}, i = 1, 2, \dots, n$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_m$ and $Y_{1}, Y_{2}, \dots, Y_{n}$ be two mutually independent random samples from populations with absolutely continuous distribution functions F_X and F_Y, respectively. For N = m + n, define $T_N = \sum(i=1..N)$ iZ$_{i}$ where $Z_{i} = 1$ if the i-th observation in the combined ordered arrangement of N observations is from F_X; and $Z_{i} = 0$, otherwise. Then, which of the following statements are true?

Consider a probability proportional to size without replacement sample involving two draws from a population of N (> 6) units with normed size measures $p_{i}'s (p_{i} > 0, i = 1, 2, \dots, N; \sum(i=1..N)p_{i} = 1)$. 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?

Suppose that a sequence of random variables {$X_{n}$}$_{n}\ge_{1}$ and the random variable X are defined on the same probability space. Then which of the following statements are true?

Let {(X_k, Y_k)}(k≥1) be a sequence of independent and identically distributed (i.i.d.) random vectors with common joint probability density function f(x, y) = e^(−y) if $0 < x < y < \infty$, and 0 otherwise. For n = 1, 2, 3, …, let $N_{n}$ be a random variable denoting the number of elements in the set {k : k = 1, 2, …, n; Y_k ≥ 2}. Then, which of the following statements are true?

Consider the Markov chain with state space {0, 1, 2} and the transition probability matrix P = [[0, 1/2, 1/2], [3/4, 0, 1/4], [3/4, 1/4, 0]]. Let P⁽$^{n}$⁾ = ((P⁽$^{n}$⁾$_{ij}))$ denote the n-step transition probability matrix. Then, which of the following statements are true?

Let $Z_{1}$ and $Z_{2}$ be two independent discrete random variables such that $Z_{1}$ follows binomial distribution with parameters n = 2 and p = 1/2, and $Z_{2}$ follows Poisson distribution with mean 1. Consider the following system of equations with three variables $x_{1}, x_{2}$ and $x_{3}: x_{1} - 2x_{2} + x_{3} = 1; 2x_{1} - 5x_{2} + 2x_{3} = 2; x_{1} + 2x_{2} + Z_{1}x_{3} = Z_{2}$. Then, the probability that the given system of equations has infinite number of solutions equals

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from an exponential distribution with probability density function f(x|$\theta) = e^(-(x-\theta))$ if $x \ge \theta$, and 0 otherwise, where $\theta \in \mathbb{R}$ is an unknown parameter. If $X_{(1)} = \min${$X_{1}, X_{2}, \dots, X_{n}$}, 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?

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from $N(\theta, 1)$ distribution, where $\theta \in \mathbb{R}$ is unknown. Let $\delta_{n}$ be the Bayes estimator of $\theta$, under the squared error loss function $L(\theta, a) = (a - \theta)^{2}, a, \theta \in \mathbb{R}$ and the prior distribution N(1, 2). If $(1/\sqrt{n})[(2n + 1)\delta_{n} - 1 - 2n\theta]$ converges in distribution to a random variable Z, as $n \to \infty$, then which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}$ be a random sample from Uniform$[0, \theta]$ distribution, $\theta > 0$. Consider the likelihood ratio test of size 0.001 for testing $H_{0} : \theta = 3$ against $H_{1} : \theta \ne 3$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}$ be a random sample of size 3 from an absolutely continuous distribution that is symmetric about 0. For i = 1, 2, 3, let $R_{i}$ denote the rank of |$X_{i}$| among |$X_{1}$|, |$X_{2}$| and |$X_{3}$|. If T⁺ $= \sum(i : X_{i}>0) R_{i}$ is the Wilcoxon signed-rank statistic, 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 $(X_{1}, X_{2})$ be a bivariate normal random vector with $E(X_{1}) = 1, E(X_{2}) = 0, Var(X_{1}) = 1, Var(X_{2}) = 1$, and correlation coefficient 1/2. Let U be a U(0, 1) random variable, which is independent of $(X_{1}, X_{2})$. If Z = (UX$_{1} + X_{2} - U)/\sqrt{U^{2} + U + 1}$, then which of the following statements are true?