CSIR NET Maths previous year question papers

Let X and Y be independent and identically distributed N(0, 1) random variables. Let $S = X^{2} + Y^{2}$ and T = $e^{-(X^{2}+Y^{2})/2}$. 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?

Consider the Markov chain with state space {0, 1, 2} and the transition probability matrix P = [[0, 1/2, 1/2], [3/4, 0, 1/4], [3/4, 1/4, 0]]. Let P⁽$^{n}$⁾ = ((P⁽$^{n}$⁾$_{ij}))$ denote the n-step transition probability matrix. 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 be a random variable with probability density function $f(x) = 1/(\pi(1 + x^{2})), x \in \mathbb{R}$. If $Z = 1/2 + (1/\pi)\tan$⁻$^{1}(X)$, then which of the following statements are true?

Transition probability matrix of a homogeneous Markov chain with states 0, 1, 2, 3 is P = [[1/4, 3/4, 0, 0], [1, 0, 0, 0], [2/3, 0, 1/3, 0], [0, 0, 2/5, 3/5]], where rows and columns are indexed 0, 1, 2, 3. Which of the following statements are true?

Let $X_{1}, X_{2}$ denote lifetimes (in years) of 2 components of an electronic system. Let $Y_{1} = X_{1} + X_{2}, Y_{2} = \max${$X_{1}, X_{2}$} and $Y_{3} = \min${$X_{1}, X_{2}$}. Assume that $X_{1}$ and $X_{2}$ are independent, each following exponential distribution with probability density function f(x) = (1/2)e^(−x/2) if x > 0, and 0 otherwise. Which of the following statements are true?

Let $X_{1}, X_{2}, X_{3}$ be a random sample from a continuous distribution having cumulative distribution function F(t), probability density function f(t), and failure rate function r(t) = f(t)/(1 − F(t)), t > 0, where F(0) = 0. If r(t) = 1 for all t > 0, then which of the following statements are true?