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

Suppose the distribution of X given $\theta$ is normal with mean $\theta$ and variance 15. Further, let the prior (improper) distribution of $\theta$ be proportional to $1, -\infty < \theta < \infty$. If the observed value of X is 13, then which of the following statements is true?

Let $Z_{1}, Z_{2}, \dots$ be a sequence of independent and identically distributed random variables having discrete uniform distribution over {1, 2, …, 2024}. Let $Y_{n} = \sum_{i}$₌$_{1}^{n} Z_{i}, n \ge 2$. Further, let $X_{n}$ be the remainder when $Y_{n}$ is divided by 2025. Then, which of the following statements is true?

A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function f(x) = $e^{-x}$ if x > 0 and 0 otherwise, and that of Brand II follows a gamma distribution with the probability density function g(x) = $(x/4)e^{-x/2}$ if x > 0 and 0 otherwise. The company uses the batteries of Brands I and II in proportion of 20% and 80% respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more that 2 years is

Consider a discrete random variable X with the probability mass function $P(X = 0) = \theta/3, P(X = 1) = 1 - \theta/2, P(X = 2) = \theta/6$, where $\theta \in (0, 1)$ is an unknown parameter. In a random sample of size 90 from this distribution, the observed counts for X = 0, 1 and 2 are 20, 60 and 10, respectively. Then, the maximum likelihood estimate of $\theta$ is

Let X be a random sample of size 1 from the probability density function f(x|$\theta) = (3/\theta^{3})(\theta - x)^{2}$ if $0 < x < \theta$, and 0 otherwise. If $(X/(1 - \lambda_{1}), X/(1 - \lambda_{2}))$ is a confidence interval for $\theta$ with confidence coefficient $1 - \alpha$, where $\lambda_{i} \in (0, 1), i = 1, 2, \lambda_{1} < \lambda_{2}$, and $\alpha \in (0, 1)$, then which of the following statements is true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from a continuous distribution with the common probability density function f(x|$\theta) = 2^\theta \theta /$ $x^{\theta+1}$ if x > 2, and 0 otherwise, where $\theta (> 0)$ is an unknown parameter. Suppose $P(Y > \chi^{2}_{m,\beta})$ $= \beta$, where Y ~ $\chi^{2}_m$. For testing $H_{0} : \theta = 1$ against $H_{1} : \theta > 1, a$ uniformly most powerful test of size $\alpha, 0 < \alpha < 1$, will reject $H_{0}$ if

Consider the multiple linear regression model $y_{i} = \beta_{0} + \beta_{1}x_{1i} + \dots + \beta_{8}x_{8i} + \varepsilon_{i}, i = 1, 2, \dots, 29$, where $\varepsilon_{1}, \varepsilon_{2}, \dots, \varepsilon_{29}$ are independent and identically normal distributed with mean 0 and variance $\sigma^{2}$. Suppose the model is fitted using the method of least squares. If the calculated value of the F-statistic for testing the significance of regression is 2.50, then the possible values of $R^{2}$ and Adjusted $R^{2}$ are respectively

Suppose $(X_{1}, X_{2}, X_{3})^{T}$ ~ $N_{3}(0, I_{3})$, where 0 is the zero mean vector and $I_{3}$ is the 3 × 3 identity matrix, and $(X, Y, Z)^{T} = A(X_{1}, X_{2}, X_{3})^{T}$ where A has rows (3, 0, 0), (2, 2, 0) and (4, 0, 4). Then the partial correlation coefficient $\rho_{YZ.X}$ is

Suppose we want to estimate the population mean Ȳ of a variable for a finite population of size 85, with 34 Statisticians and 51 Biologists. We consider the following sampling scheme: a stratified random sample with 2 strata of Statisticians (Stratum-1) and Biologists (Stratum-2), where 12 Statisticians and 15 Biologists are drawn from Stratum-1 and Stratum-2, respectively, using SRSWOR scheme. Denote ȳ_S, ȳ_B, and ȳ as the mean of the variable among the Statistician sample, Biologist sample, and the combined sample, respectively. Which of the following is an unbiased estimator of Ȳ?

Solve the following linear programming problem: maximize z = x + y subject to 5x + 3y ≤ 30, 2x + 6y ≤ 25, 2x − y ≤ 8, x ≥ 0, y ≥ 0. Then the optimal value of the objective function is

Let X be a Binomial(n, p) random variable, where n ∈ {5, 6} and p ∈ {1/4, 3/4}. If X = 3 is observed, then the maximum likelihood estimate of (n, p) is

Let {$X_{n} : n \ge 0$} be any homogeneous Markov Chain on the state space S = {1,2,3,4} having the transition probability matrix $P = (p_{ij})$ given by row 1 = (1/4, 0, 2/3, 1/12), row 2 = (0, 1, 0, 0), row 3 = (1/12, 0, 1/4, 2/3), row 4 = (2/3, 0, 1/12, 1/4). Which of the following statements about stationary distributions of any such Markov Chain is true?

Let X, Y, and Z be independent Normal random variables with means −1, 0, and 1, respectively, and variances 1, 1, and 3, respectively. Which of the following random variables has a Cauchy distribution with location parameter 0 and scale parameter 1?

Consider the following linear programming problem: Maximize $3x_{1} + 4x_{2}$ subject to $3x_{1} + 2x_{2} \le 12, 3x_{1} + 5x_{2} \le 15, 2x_{1} - x_{2} \ge 0, x_{2} \le 2$, and $x_{1}, x_{2} \ge 0$. Which of the following values is the optimum value of the objective function in the feasible region?

Let $X = (X_{1},X_{2},X_{3})^{T}$ be a 3×1 random vector with E(X) = (3,2,1)ᵀ and $Cov(X) = \sum$, with rows (3,−2,0), (−2,3,−2), (0,−2,3). Suppose that $Y = (Y_{1},Y_{2},Y_{3})^{T} = (1/\sqrt{3})X$. Then the value of the multiple correlation coefficient between $Y_{1}$ and $(Y_{2},Y_{3})$ equals

Consider a finite population of size N = 100. Let $T_{1}$ be the sample mean of a study variable based on a sample of size n (1 < n < N) under simple random sampling with replacement scheme. Let $T_{2}$ be the sample mean of the same study variable based on a sample of size n under simple random sampling without replacement scheme. If $Var(T_{1}) = 9Var(T_{2})$, then the sample size n equals

Let $X_{1}$ and $X_{2}$ be a random sample from Uniform$[0,\theta]$ distribution, where $\theta > 0$. For testing the hypothesis $H_{0}: \theta=1$ against $H_{1}: \theta=2$, consider a test which rejects $H_{0}$ if $X_{1}+X_{2} > 4/5$. Then, the probability of type-I error 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 single sample from an absolutely continuous distribution with probability density function f(x∣$\theta) = (2/\theta^{2})(\theta-x)$ if $0 < x < \theta$, and 0 otherwise, where $\theta > 0$ is unknown. Which of the following intervals is a 95% confidence interval for $\theta$?

Consider the linear model $Y_{1} = \beta_{1}+\beta_{2}+\varepsilon_{1}, Y_{2} = \beta_{1}+2\beta_{2}+\varepsilon_{2}, Y_{3} = \beta_{1}+c\beta_{2}+\varepsilon_{3}$, where $\beta_{1},\beta_{2} \in \mathbb{R}$ are unknown parameters, and the uncorrelated errors $\varepsilon_{i}, i=1,2,3$ have zero mean and finite variance $\sigma^{2}(>0)$. The constant c is such that $\beta$̂$_{1}$ and $\beta$̂$_{2}$ are uncorrelated, where $\beta$̂$_{1}$ and $\beta$̂$_{2}$ are the best linear unbiased estimators of $\beta_{1}$ and $\beta_{2}$, respectively. Which of the following statements is the correct option for $(Var(\beta$̂$_{1}), Var(\beta$̂$_{2}))$?

Let $X_{1}, X_{2}, \dots$ be a sequence of independent and identically distributed random variables having Binomial(3, 1/4) distribution. For j=1,2,…, define $Y_{j} = 1$ if $X_{j} \le \sqrt{5}$, and 0 otherwise. If $(1/n)\sum_{j=1}^n$ $Y_{j}^{2}$ converges almost surely to a constant c as $n \to \infty$, then the value of c equals

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 a random sample from $N(\theta, 1), \theta \in \mathbb{R}$. If $\theta$̂ is the Bayes estimator of $\theta$ with respect to some prior $\pi(\theta)$ and loss function $L(\theta, d)$. Then, which of the following statements are true?