CSIR NET Maths previous year question papers

Let $X_{1}, X_{2}, \dots$ be a sequence of independent and identically distributed random variables with $E(X_{1}) = 0, E(X_{1}^{2}) = 1, E(X_{1}^{3}) = 0, E(X_{1}^{4}) = 3$. Let $S_{n} = \sum_{i}$₌$_{1}^{n} X_{i}, T_{n} = \sum_{i}$₌$_{1}^{n} X_{i}^{2}, U_{n} = \sum_{i}$₌$_{1}^{n} X_{i}^{3}$ and $V_{n} = \sum_{i}$₌$_{1}^{n} X_{i}^{4}$. Then, which of the following statements are true?

Let {$X_{n} : n\ge1$} be a sequence of independent and identically distributed random variables, where $X_{1}$ has an Exponential distribution with mean 1. Define $T_{n} = \max${$X_{1},X_{2},\dots,X_{n}$} − ln n, n≥1. Suppose $T_{n} \to$ᵈ Y as $n\to\infty$. 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?

Let $\mu$ denote the Lebesgue measure on $\mathbb{R}$ and $f : \mathbb{R} \to [0, \infty)$ be a Lebesgue measurable function. For n ≥ 1, let $E_{n} =$ {$x \in \mathbb{R} : n - 1 \le f(x) < n$}. Suppose that $\sum(n\ge1) n^{2}\mu(E_{n}) < \infty$. Which of the following statements are true?

Suppose U ~ Uniform(0,1) and $X = \tan(\pi(U -$ ½)). Then which of the following statements are true?

Let $X_{1}, \dots, X_{n}$ be a random sample from an unknown absolutely continuous CDF F, and $F_{0} a$ specified absolutely continuous CDF. For $H_{0}: F = F_{0}$ vs $H_{1}: F \ne F_{0}$, consider $T_{1,n}$ = sup_x |$F_{n}(x) - F_{0}(x)$| and $T_{2,n}$ = n·sup_x |$F_{n}(x) - F_{0}(x)$|, where $F_{n}$ is the empirical CDF. Which of the following statements are true?