CSIR NET Maths previous year question papers

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

Let $X_{1}, X_{2}, X_{3}, X_{4}$ be i.i.d. Bernoulli(1/3) and $X_{(1)} \le \cdots \le X_{(4)}$ the order statistics. Which of the following is true?

Let X be a Poisson random variable with mean $\lambda$. Which of the following parametric functions is not estimable?

Let $X_{1}, \dots, X_{7}$ and $Y_{1}, \dots, Y_{9}$ be independent random samples from continuous CDFs F and G. For the Wald–Wolfowitz run test of $H_{0}: F = G$ vs $H_{1}: F \ne G$, let R be the total number of runs in the combined ordered arrangement. Which of the following is true?

Consider the simple linear regression model $Y_{i} = \beta{}x_{i} + \varepsilon_{i}, i = 1, \dots, n$, where $E(\varepsilon_{i}) = 0, Cov(\varepsilon_{i}, \varepsilon_{k}) = 0$ for i ≠ k and $Var(\varepsilon_{i}) = x_{i}^{2}\sigma^{2}$. The best linear unbiased estimator of $\beta$ is

Let $X = (X_{1}, X_{2})^{T}$ be bivariate normal with mean (0, 0)ᵀ and covariance $\sum = [[5, -3], [-3, 10]]$. The mean vector and covariance matrix of $Y = (X_{1}, 5 - 2X_{2})^{T}$ are

Suppose $X = (X_{1}, X_{2}, X_{3}, X_{4})^{T}$ ~ $N_{4}(0, I_{2}$ ⊗ $\sum)$ with $\sum = [[2, -1], [-1, 2]]$. Define $Z = [[X_{1}, X_{2}], [X_{3}, X_{4}]]$ and Q = ZᵀZ. If $W_{m}(n, \sum)$ denotes a Wishart distribution of order m with n degrees of freedom, the distribution of $Q_{11} + Q_{12} + Q_{21} + Q_{22}$ is

Suppose X ~ Poisson(3/4). Then which of the following statements is true?

Let (X, Y) have joint moment generating function $M(t_{1}, t_{2}) = (3/4 +$ $e^{t_{1}}/4)^{2}$ (5/6 + $e^{t_{2}}/6)^{3}$. Then P(X + 2Y > 1) equals

Let $X_{1}, X_{2}, \dots$ be i.i.d. with CDF F(x) = 0 for x < 5 and 1 − $e^{5-x}$ for x ≥ 5. Define $Y_{n} = \min${$X_{1}, \dots, X_{n}$} and $Z_{n} = \sqrt{n} (Y_{n} - 5)$, and let Z be standard normal. Which one of the following statements is true?

The probability of a head in tossing a coin is p ∈ (0, 1). The coin is independently tossed 25 times and heads appear 10 times. The Bayes estimate of p with respect to the prior Beta(5, 5) and squared error loss is

For n ≥ 2, let $X_{1}, \dots, X_{n}$ be a random sample with density f(x|$\theta) =$ $\theta{}x^{\theta-1}$ on $0 < x < 1, \theta > 0$ unknown. Which of the following is the UMVUE for $1/\theta$?

For n ≥ 2, let $\varepsilon_{1}, \dots, \varepsilon_{n}$ be $i.i.d. N(0, \sigma^{2})$ and $Y_{i} = i\alpha + i^{2}\alpha^{2} + \varepsilon_{i}$, where $\sigma > 0$ and $\alpha \in \mathbb{R}$ are unknown. Which of the following is a jointly minimal sufficient statistic for $(\alpha, \sigma)$?

Let $X_{1}, \dots, X_{6}$ be a random sample from the gamma density f(x|$\lambda) = (\lambda^{4}/6)$ $e^{-\lambda{}x}$ $x^{3}$ for $x > 0, \lambda > 0$ unknown. Let $T = \sum{}X_{i}$ and $\psi$ be the UMP test of size 0.05 for $H_{0}: \lambda = 1$ against $H_{1}: \lambda > 1$. With $\chi^{2}_{\nu,\alpha}$ the $(1 - \alpha)-$th quantile of $\chi^{2}_\nu$, the test $\psi$ rejects $H_{0}$ iff

Consider $Y = \alpha + \beta{}x + \varepsilon$ with $E(\varepsilon) = 0$. From 10 observations the OLS fit is ŷ = 1.5 + 0.8x. Suppose $\sum(y_{i} -$ ȳ$)^{2} = 5$ and $\sum(x_{i} - {\bar{x}})^{2} = 6$. The adjusted $R^{2} ($to two decimals) equals

For n ≥ p + 1, let $X_{1}, \dots, X_{n}$ be a random sample from $N_{p}(\mu, \sum)$ with $\sum$ positive definite. Define X̄ and $A = \sum(X_{i} - {\bar{X}})(X_{i} - {\bar{X}})^{T}$. The distribution of Trace$(A\sum$⁻$^{1})$ is

In a Latin square design, the degrees of freedom for the error sum of squares is 42. Then the degrees of freedom for the sum of squares due to treatments is