CSIR NET Maths previous year question papers

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?

The probability of a head in tossing a coin is p ∈ (0, 1). The coin is independently tossed 25 times and heads appear 10 times. The Bayes estimate of p with respect to the prior Beta(5, 5) and squared error loss is

For n ≥ 2, let $X_{1}, \dots, X_{n}$ be a random sample with density f(x|$\theta) =$ $\theta{}x^{\theta-1}$ on $0 < x < 1, \theta > 0$ unknown. Which of the following is the UMVUE for $1/\theta$?

For n ≥ 2, let $\varepsilon_{1}, \dots, \varepsilon_{n}$ be $i.i.d. N(0, \sigma^{2})$ and $Y_{i} = i\alpha + i^{2}\alpha^{2} + \varepsilon_{i}$, where $\sigma > 0$ and $\alpha \in \mathbb{R}$ are unknown. Which of the following is a jointly minimal sufficient statistic for $(\alpha, \sigma)$?

Let $X_{1}, \dots, X_{6}$ be a random sample from the gamma density f(x|$\lambda) = (\lambda^{4}/6)$ $e^{-\lambda{}x}$ $x^{3}$ for $x > 0, \lambda > 0$ unknown. Let $T = \sum{}X_{i}$ and $\psi$ be the UMP test of size 0.05 for $H_{0}: \lambda = 1$ against $H_{1}: \lambda > 1$. With $\chi^{2}_{\nu,\alpha}$ the $(1 - \alpha)-$th quantile of $\chi^{2}_\nu$, the test $\psi$ rejects $H_{0}$ iff

Consider $Y = \alpha + \beta{}x + \varepsilon$ with $E(\varepsilon) = 0$. From 10 observations the OLS fit is ŷ = 1.5 + 0.8x. Suppose $\sum(y_{i} -$ ȳ$)^{2} = 5$ and $\sum(x_{i} - {\bar{x}})^{2} = 6$. The adjusted $R^{2} ($to two decimals) equals

For n ≥ p + 1, let $X_{1}, \dots, X_{n}$ be a random sample from $N_{p}(\mu, \sum)$ with $\sum$ positive definite. Define X̄ and $A = \sum(X_{i} - {\bar{X}})(X_{i} - {\bar{X}})^{T}$. The distribution of Trace$(A\sum$⁻$^{1})$ is

In a Latin square design, the degrees of freedom for the error sum of squares is 42. Then the degrees of freedom for the sum of squares due to treatments is