CSIR NET Maths previous year question papers

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 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 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 $\theta$?

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 3)$ be a random sample from the uniform distribution on the interval $(\theta_{1} - \theta_{2}, \theta_{1} + \theta_{2})$, where $\theta_{1} \in \mathbb{R}$ and $\theta_{2} > 0$ are unknown parameters. Let $X_{(j)}$ be the jᵗʰ order statistic, j = 1, 2, …, n, and let ${\bar{X}} = (1/n)\sum_{i}$₌$_{1}^{n} X_{i}$. Here, $(X_{(1)}, X_{(n)})$ is a complete and sufficient statistic for $(\theta_{1}, \theta_{2})$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 2)$ be a random sample from a continuous distribution with the probability density function f(x|$\theta) =$ $(1/2\theta)e^{-|x-\theta|/\theta}$, $x \in \mathbb{R}$, where $\theta (> 0)$ is an unknown parameter. Let $U_{n} = (1/n)\sum_{i}$₌$_{1}^{n} X_{i}, V_{n} = (1/n)\sum_{i}$₌$_{1}^{n} X_{i}^{2}$, and $S_{n}^{2} = (1/(n-1))\sum_{i}$₌$_{1}^{n}(X_{i} - U_{n})^{2}$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n} (n\ge2)$ be a random sample from an Exponential distribution with the probability density function f(x∣$\mu,\sigma) = (1/\sigma)\exp((\mu-x)/\sigma)$ if $x>\mu, 0$ otherwise, where parameters $\mu$ and $\sigma$ are unknown and positive. Let ${\bar{X}}_{n}, S_{n}^{2}$ and $X_{1}:_{n}$ denote the sample mean, the sample variance and the sample smallest order statistic, respectively, and let $\theta = \sigma/\mu$. 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?

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 X and Y be independent random variables such that X follows U(0, 1) distribution and Y follows Bernoulli distribution with success probability p ∈ (0, 1). Define Z = X + Y. Let $z_{1} = 0.5, z_{2} = 1.2, z_{3} = 1.3, z_{4} = 0.9, z_{5} = 0.1, z_{6} = 0.7$ be the observed values from the distribution of Z. Then the maximum likelihood estimate of p equals

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

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from distribution Poisson$(\theta), \theta > 0$. Let $\pi(\theta) = e^(-\theta), \theta > 0$, be the prior distribution of $\theta$. Under the squared error loss function, which of the following is the Bayes estimator of $e^(-2\theta)$?

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

Consider a distribution with probability mass function f(x | $\theta) = (1 - \theta)/2$ if x = 0; 1/2 if $x = 1; \theta/2$ if x = 2; and 0 otherwise, where $\theta \in (0, 1)$ is an unknown parameter. In a random sample of size 100 from the above distribution, the observed counts of 0, 1 and 2 are 20, 30 and 50 respectively. Then, the maximum likelihood estimate of $\theta$ based on the observed data is

Let $X_{1}, \dots, X_{10}$ be a random sample from a distribution with the probability density function f(x|$\theta) = \theta{}x^(\theta-1)$ if 0 < x < 1, and 0 otherwise, where $\theta > 0$ is an unknown parameter. The prior distribution of $\theta$ is given by $\pi(\theta) = \theta{}e^(-\theta)$ if $\theta > 0$, and 0 otherwise. The Bayes estimator of $\theta$ under squared error loss is

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?

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

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 $X_{1}, \dots, X_{n}$ 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?