CSIR NET Maths previous year question papers

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 {$X_{n}$ | n ≥ 0} be a homogeneous Markov chain with state space S = {0, 1, 2, 3, 4} and transition probability matrix

Let $\alpha$ denote the probability that starting with state 4 the chain will eventually get absorbed in closed class {0, 3}. Then the value of $\alpha$ is

Consider a petrol pump which has a single petrol dispensing unit. Customers arrive there in accordance with a Poisson process having rate $\lambda = 1$ minutes. An arriving customer enters the petrol pump only if there are two or less customers in the petrol pump, otherwise he/she leaves the petrol pump without taking the petrol (at any point of time a maximum of three customers are present in the petrol pump). Successive service times of the petrol dispensing unit are independent exponential random variables having mean 1/2 minutes. Let X denote the average number of customers in the petrol pump in the long run. Then E(X) is equal to

Let a point P be chosen at random on the line segment AB of length $\alpha$. Let $Z_{1}$ and $Z_{2}$ denote the lengths of line segments AP and BP respectively. Then the value of E(|$Z_{1} - Z_{2}$|) 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 $X_{1}, X_{2}$ be a random sample from $N(0, \sigma^{2})$ distribution, where $\sigma > 0$ and $N(\mu, \sigma^{2})$ denotes a normal distribution with mean $\mu$ and variance $\sigma^{2}$. Suppose, for some constant $c, (c(X_{1}^{2} + X_{2}^{2}), \infty)$ is a confidence interval for variance $\sigma^{2}$ with confidence coefficient 0.95. Then the value of c 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

An analyst considers standardized values of observations on three variables, consumption (C), saving (S) and total income (TI) so that they have zero means and unit variances. She further considers disposable income (DI) where DI = C + S. In the simple linear regressions of DI on TI, DI on C and S on TI, the regression coefficients are 0.8, 0.5 and 0.4, respectively. There are 21 sample observations. Sample covariances and variances are calculated with divisor 20. Then, the value of sum of squared residuals in the regression of DI on S is

Let $X_{0}, X_{1}, \dots, X_p (p \ge 2)$ be independent and identically distributed random variables with mean 0 and variance 1. Suppose $Y_{i} = X_{0} + X_{i}, i = 1, \dots, p$. The first principal component based on the covariance matrix of $Y = (Y_{1}, \dots, Y_p)^{T}$ is

The expected number of distinct units in a simple random sample of 3 units drawn with replacement from a population of 100 units is

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?