CSIR NET Maths previous year question papers

Consider the following $2^{5}-$factorial design with 8 blocks (Block 1: (1), acd, bce, abde; Block 2: e, acde, bc, abd; and six further blocks). Which of the following statements are true?

Let {$X_{n} : n\ge1$} be a sequence of independent and identically distributed random variables, where $X_{1}$ has an Exponential distribution with mean 1. Define $T_{n} = \max${$X_{1},X_{2},\dots,X_{n}$} − ln n, n≥1. Suppose $T_{n} \to$ᵈ Y as $n\to\infty$. Then which of the following statements are true?

Suppose that the transition probability matrix of a homogeneous Markov chain with state space {1,2,3,4} is given by P, with row 1 = (1/4, 3/4, 0, 0), row 2 = (1, 0, 0, 0), row 3 = (1/8, 0, 7/8, 0), row 4 = (0, 0, 1/9, 8/9). Which of the following statements are true?

Let X and Y be two independent random variables such that the moment generating functions of X and Y are $M_X(t)=e^(3(e^t-1)), t\in\mathbb{R}$, and $M_Y(t)=((1/2)e^(-3t)+(1/2)e^(3t))^{2}, t\in\mathbb{R}$, respectively. Then which of the following statements are true?

Let $X_{1},X_{2},\dots,X_{7}$ be a random sample drawn from a continuous distribution with unknown unique median M. The null hypothesis $H_{0}:M=2$ is tested against the alternative $H_{1}:M>2$ at level of significance 0.05 using the right-tailed test based on the Sign test statistic K, which is the number of observations in the sample greater than 2. If the observed sample is −3,−6,1,9,4,10,12, which of the following statements are true?

Suppose two fair dice are thrown independently at random. Let X and Y be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?

Consider a series system comprising four components, having independent and identically distributed lifetimes with hazard rate $\lambda(t)=1/(1+t), t>0$, and survival function S(t), t>0. If Y denotes the lifetime of the series system, then which of the following statements are true?

Let $X_{1},X_{2},\dots,X_{n} (n>5)$ be independent random variables such that $X_{t}=\alpha+\alpha^{2}t+\varepsilon_{t}$, for t=1,…,n, where $\varepsilon_{1},\varepsilon_{2},\dots,\varepsilon_{n}$ are independent and identically distributed $N(0,\sigma^{2})$ random variables. Here $\alpha\in\mathbb{R}$ and $\sigma>0$ are unknown parameters. Which of the following statements are true?

Let $Y=X\beta+\varepsilon$ be a multiple linear regression model with p regressors and an intercept, where $\beta=(\beta_{0},\beta_{1},\dots,\beta_{p})^{T}$ and the random error $\varepsilon$~$N_{n}(0,\sigma^{2}I), \sigma>0$ and n>p+1. The least squares method provides a unique estimator $\beta$̂. Let the total sum of squares (corrected), sum of squares due to regression, and sum of squares due to error, based on $\beta$̂, be denoted YᵀAY, YᵀBY and YᵀCY respectively, so YᵀAY=YᵀBY+YᵀCY. Which of the following statements are always true?

Consider the simple linear regression model $Y_{i}=\beta{}x_{i}+\varepsilon_{i}, i=1,2,\dots,n$, where $\beta>0, \sum{}x_{i}^{2}>0$, and the uncorrelated errors $\varepsilon_{i}$ have zero mean and finite variance $\sigma^{2}(>0)$. Let $\beta$̃$_{1}=\sum{}a_{i}$$Y_{i}$, where $a_{i}$'s minimize $E(\sum{}a_{i} Y_{i}-\beta)^{2}$ with respect to scalars $a_{1},a_{2},\dots,a_{n}$. Let $\beta$̃$_{2}$ be the ordinary least squares estimator of $\beta$. Which of the following statements are true?

Consider maximizing the objective function $P = x_{1} + 2x_{2} + 3x_{3}$ subject to $x_{1} + x_{2} \le 5, x_{2} + x_{3} \le 2, x_{1} + x_{2} + x_{3} \le 6, x_{1} \ge 0, x_{2} \ge 0, x_{3} \ge 0$. Then, which of the following statements are true?

Let $X_{1}, X_{2} \dots$ be a sequence of independent random variables with $X_{i}$ following N(0, 1 + 1/i) distribution for all $i \in \mathbb{N}$. Let X be N(0, 1)-random variable independent of {$X_{n} : n \ge 1$}. Then, which of the following statements are true?

Let $U_{1}, U_{2}, \dots$ be a sequence of independent and identically distributed (i.i.d.) U(0, 1) random variables. Let $G_{n} = (\prod(i=1..n) U_{i})^(1/n)$ be the geometric mean of $U_{1}, U_{2}, \dots, U_{n}$ for $n \in \mathbb{N}$. Let $X_{1}$ and $X_{2}$ be degenerate random variables such that $P(X_{1} = 0) = 1$ and $P(X_{2} = 1/e) = 1$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed (i.i.d.) random variables such that for $x = 3, 4, \dots, P(X_{1} = \pm{}x) = 1/(2$cx$^{2}\ln(x))$, where $c = \sum(x=3..\infty) 1/(x^{2}\ln(x))$. Then, which of the following statements are true?

Consider the Markov chain with state space {0, 1, 2, 3, 4} and the transition probability matrix P = [[0.4, 0.3, 0.3, 0, 0], [0, 0.5, 0, 0.5, 0], [0.5, 0, 0.5, 0, 0], [0, 0.5, 0, 0.5, 0], [0, 0.3, 0, 0.3, 0.4]]. Then, which of the following statements are true?

Let X and Y be independent random variables with the moment generating functions M_X(t) = (1/9)(2 + eᵗ$)^{2}, t \in \mathbb{R}$ and M_Y(t) = exp[eᵗ $- 1], t \in \mathbb{R}$, respectively. 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 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?