CSIR NET Maths previous year question papers

A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function f(x) = $e^{-x}$ if x > 0 and 0 otherwise, and that of Brand II follows a gamma distribution with the probability density function g(x) = $(x/4)e^{-x/2}$ if x > 0 and 0 otherwise. The company uses the batteries of Brands I and II in proportion of 20% and 80% respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more that 2 years is

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?

Suppose that {$X_{n}$}$(n\in\mathbb{N})$ is a sequence of independent and identically distributed (i.i.d.) random variables with the common probability density function $f(x) = 1/(\pi(1 + x^{2})), x \in \mathbb{R}$. Then which of the following statements is 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 (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?

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?