CSIR NET Maths previous year question papers

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

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 paired data $(x_{i}, y_{i}); i = 1, 2, 3, 4, 5$, where $(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) = (-2, -1, 0, 1, 2)$ and $y_{i} = x_{i}^{2}$ for all i = 1, 2, 3, 4, 5. On this data, a simple linear regression model with an intercept term and a simple linear regression model without an intercept term are fitted using the method of least squares. Which of the following statements are true?

Let X and Y be two independent random variables such that the moment generating functions of X and Y are $M_X(t)=e^(3(e^t-1)), t\in\mathbb{R}$, and $M_Y(t)=((1/2)e^(-3t)+(1/2)e^(3t))^{2}, t\in\mathbb{R}$, respectively. Then which of the following statements are true?

Suppose two fair dice are thrown independently at random. Let X and Y be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?

Planes take off in a busy airport in accordance with the Poisson process with rate 60 planes per hour. 10% of these planes are cargo planes and 90% are passenger planes. Given that 10 cargo planes have taken off during one hour, what is the expected total number of planes that have taken off in that hour?

Consider a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1)$, where errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Here, $Y_{i}$ is the i-th response. Let Ŷ$_{i}$ be the i-th predicted response by the least squares estimation method, and let $\varepsilon$̂$_{i} = Y_{i} -$ Ŷ$_{i}, 1 \le i \le n$. Then, which of the following statements is true?

Let X, Y and Z be independent and identically distributed (i.i.d.) random variables with distribution N(0, 1). Define U = 2X, V = 3X + Y, W = X + 4Z. Then the partial correlation coefficient of V and W, given U is

Consider a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1)$, where the errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Here, $Y_{i}$ is the i-th response. Let Ŷ$_{i}$ be the i-th predicted response by the least squares estimation method, and let $\varepsilon$̂$_{i} = Y_{i} -$ Ŷ$_{i}, 1 \le i \le n$. Then, which of the following statements is true?

Let $A_{1}, A_{2}, A_{3}$ be events satisfying $0 < P(A_{i}) < 1$ for i = 1, 2, 3. Which of the following statements is true?

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

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 $(X_{1}, X_{2})$ be a bivariate normal random vector with $E(X_{1}) = 1, E(X_{2}) = 0, Var(X_{1}) = 1, Var(X_{2}) = 1$, and correlation coefficient 1/2. Let U be a U(0, 1) random variable, which is independent of $(X_{1}, X_{2})$. If Z = (UX$_{1} + X_{2} - U)/\sqrt{U^{2} + U + 1}$, then which of the following statements are true?

Let $X_{1}$ and $X_{2}$ be random variables having absolutely continuous distribution functions. Let $h_{i}(t)$ denote the hazard function of $X_{i}, i = 1, 2$. If $h_{1}(t) \le h_{2}(t)$ for all $t \in \mathbb{R}$, 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 and Y be jointly distributed continuous random variables with joint probability density function f(x, y) = x/y if 0 < x < y < 2, and 0 otherwise. 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?

Let A, B be two events in a discrete probability space with P(A) > 0 and P(B) > 0. Which of the following are necessarily true?

Let $X_{i} (i = 1, 2)$ be independent with densities $f_{i}(x) = i$ $e^{-ix}$ for x ≥ 0. Consider a series system of two independent components with lifetimes $X_{1}$ and $X_{2}$, and let X denote the lifetime of the system. Which of the following statements are true?

Suppose {X(t)} and {Y(t)} are two independent homogeneous Poisson processes with the same rate $\lambda = 2$. Let $W_{n}$ˣ and $W_{n}$ʸ be the waiting times for the n-th arrival in each process. Which of the following statements are true?