CSIR NET Maths previous year question papers

Consider an M/M/2 queuing system with the birth rate $\lambda=4$ per minute, the death rate $\mu=1$ per minute, and the total capacity of 3 customers (including the ones that are being served). Let $p_{0}$ and $p_{3}$ denote the long-run probabilities that the system will be empty (i.e. without customers) and will be blocked (i.e. full), respectively. 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 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?

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?

Consider the linear programming problem: max{$x_{1} + x_{2} + x_{3}$} subject to constraints $x_{1} + x_{2} - x_{3} \le 1, x_{1} + x_{3} \le 2, 0 \le x_{1} \le 1/2, x_{2} \ge 0$, and $0 \le x_{3} \le 1$. Which of the following statements are true?

Let {$X_{i} : 1 \le i \le 2n$} be i.i.d. normal with mean $\mu$ and variance 1, independent of a standard Cauchy random variable W. Which of the following statistics are consistent for $\mu$?

Consider the one-way fixed effects ANOVA model $Y_{ij} = \mu + \alpha_{i} + \varepsilon_{ij}, j = 1, \dots, n_{i}; i = 1, \dots, k$, with uncorrelated errors of mean 0 and variance $\sigma^{2}$. Let Ȳ$_{i}$ be the i-th group mean. Which of the following statements are true?

Let $(X_{1}, Y_{1}), (X_{2}, Y_{2}), (X_{3}, Y_{3})$ be i.i.d. bivariate normal with mean (0, 0) and correlation matrix $[[1, \rho], [\rho, 1]]$, |$\rho$| < 1. Let $S_\rho = 3E($sgn$(X_{1} - X_{2})(Y_{1} - Y_{3}))$. Which of the following statements are true?