CSIR NET Maths previous year question papers

Let $X_{1}, \dots, X_{7}$ and $Y_{1}, \dots, Y_{9}$ be independent random samples from continuous CDFs F and G. For the Wald–Wolfowitz run test of $H_{0}: F = G$ vs $H_{1}: F \ne G$, let R be the total number of runs in the combined ordered arrangement. Which of the following is true?

Consider the simple linear regression model $Y_{i} = \beta{}x_{i} + \varepsilon_{i}, i = 1, \dots, n$, where $E(\varepsilon_{i}) = 0, Cov(\varepsilon_{i}, \varepsilon_{k}) = 0$ for i ≠ k and $Var(\varepsilon_{i}) = x_{i}^{2}\sigma^{2}$. The best linear unbiased estimator of $\beta$ is

Let $X = (X_{1}, X_{2})^{T}$ be bivariate normal with mean (0, 0)ᵀ and covariance $\sum = [[5, -3], [-3, 10]]$. The mean vector and covariance matrix of $Y = (X_{1}, 5 - 2X_{2})^{T}$ are

Suppose $X = (X_{1}, X_{2}, X_{3}, X_{4})^{T}$ ~ $N_{4}(0, I_{2}$ ⊗ $\sum)$ with $\sum = [[2, -1], [-1, 2]]$. Define $Z = [[X_{1}, X_{2}], [X_{3}, X_{4}]]$ and Q = ZᵀZ. If $W_{m}(n, \sum)$ denotes a Wishart distribution of order m with n degrees of freedom, the distribution of $Q_{11} + Q_{12} + Q_{21} + Q_{22}$ is

Suppose X ~ Poisson(3/4). Then which of the following statements is 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?

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

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?

Suppose $X_{1}, X_{2}, \dots$ are independent and identically distributed N(0,1) random variables and $Y_{n} = X_{1}^{4} + X_{2}^{4} + \cdots + X_{n}^{4}$. Which of the following probabilities converge to 1/2 as $n \to \infty$?

Let $X_{1}$ and $X_{2}$ be independent, $X_{1}$ gamma with mean 10 and variance 10, and $X_{2}$ ~ N(3, 4). Let $f_{1}, f_{2}$ be their densities. Define Y with density $f(y) = 0.4 f_{1}(y) + q f_{2}(y)$. Which of the following are true?

Let {$X_{i} : 1 \le i \le 2n$} be i.i.d. normal with mean $\mu$ and variance 1, independent of a standard Cauchy random variable W. Which of the following statistics are consistent for $\mu$?

Under H: X ~ p with p(x) = 1/20, and under K: X ~ q with q(x) = x/210, x ∈ {1, …, 20}. Define test functions $\varphi(x) = 1$ if x ≤ 2 (else 0) and $\psi(x) = 1$ if x ≥ 19 (else 0). 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 be a discrete random variable with support S = {−1, 0, 1} and P(X = 0) = 1/3. Then which of the following statements are true?

Suppose X is a continuous random variable with density $f(x) = (1/\pi)\cdot1/(1 + (x + 1)^{2})$, and Y = X/|X| for X ≠ 0, Y = 0 for X = 0. Then which of the following statements are true?

Let X be a discrete random variable with support {0, 1, …, 25} and $P(X = x) = C(25, x)/2^{25}$. Then which of the following statements are 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?

Consider an M/M/1 queue with arrival rate $\lambda = 15$ per hour and service rate $\mu = 45$ per hour. Let N(t) be the number of customers in the system at time t, and $T_{1}, T_{2}$ the time a customer spends in the queue and in the system. Which of the following statements are true?