CSIR NET Maths previous year question papers

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 $U_{1}, U_{2}, \dots, U_{6}$ be 6 urns such that urn U_k contains $3k + k^{2}$ balls, out of which 3k are white balls and $k^{2}$ are black balls, k = 1, 2, …, 6. An urn is selected with the probability of selecting urn U_k being proportional to (k + 3). A ball is chosen randomly from the selected urn. Then the probability that urn $U_{6}$ was selected, given that the ball drawn is white, is equal to

Planes take off in a busy airport in accordance with the Poisson process with rate 60 planes per hour. 10% of these planes are cargo planes and 90% are passenger planes. Given that 10 cargo planes have taken off during one hour, what is the expected total number of planes that have taken off in that hour?

Consider an M/M/3 queuing system with arrival rate $\lambda = 2$ and service rate $\mu = 3/2$. Define, for $i = 1, 2, \dots, I_{i} = 1$ if the first transition is from i to i + 1, and $I_{i} = 0$ if the first transition from i is i − 1. Then, $Var(I_{4})$ equals

Suppose X ~ Uniform(5, 10). Define Z = X + 3 if X ≤ 7, and Z = X − 3 otherwise. Then E(Z) is

Let X and Y be independent random variables such that X follows U(0, 1) distribution and Y follows Bernoulli distribution with success probability p ∈ (0, 1). Define Z = X + Y. Let $z_{1} = 0.5, z_{2} = 1.2, z_{3} = 1.3, z_{4} = 0.9, z_{5} = 0.1, z_{6} = 0.7$ be the observed values from the distribution of Z. Then the maximum likelihood estimate of p equals

Suppose that the probability density function of the random variable X is $f(x) = 2(\theta - x)/\theta^{2}$ if $0 \le x \le \theta$, and 0 otherwise, where $\theta > 0$ is an unknown parameter. Based on a single observation X, the confidence coefficient of the confidence interval [(2/5)X, (5/2)X] for $\theta$ is

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from distribution Poisson$(\theta), \theta > 0$. Let $\pi(\theta) = e^(-\theta), \theta > 0$, be the prior distribution of $\theta$. Under the squared error loss function, which of the following is the Bayes estimator of $e^(-2\theta)$?

Let X be a random variable with the probability density function $f(x) = 2\theta{}x + 2(1 - \theta)(1 - x)$ if 0 < x < 1, and 0 otherwise, where $\theta \in [0, 1]$. Based on single observation x, the critical region of the most powerful test for testing null hypothesis $H_{0} : \theta = 1/2$ against alternative hypothesis $H_{1} : \theta = 1$, at level of significance $\alpha = 0.25$, is

Consider a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1)$, where errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Here, $Y_{i}$ is the i-th response. Let Ŷ$_{i}$ be the i-th predicted response by the least squares estimation method, and let $\varepsilon$̂$_{i} = Y_{i} -$ Ŷ$_{i}, 1 \le i \le n$. Then, which of the following statements is true?

Let X, Y and Z be independent and identically distributed (i.i.d.) random variables with distribution N(0, 1). Define U = 2X, V = 3X + Y, W = X + 4Z. Then the partial correlation coefficient of V and W, given U is

Consider the following design where the columns represent blocks and the letters represent treatments — block 1: A, B; block 2: C, D; block 3: A, C; block 4: B, D; block 5: A, D; block 6: B, C; block 7: A, E; block 8: B, E; block 9: C, E; block 10: D, E. Then, which of the following statements is NOT true?

$\lim(n\to\infty) \int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1} (x_{1}^{2} + \cdots + x_{n}^{2})/(x_{1} + \cdots + x_{n})$ dx$_{1} \dots$ dx$_{n}$ equals

Let $U_{1}, U_{2}, \dots, U_{5}$ be 5 urns such that urn U_k contains $2k + k^{2}$ balls, out of which 2k are white balls and $k^{2}$ are black balls, k = 1, 2, …, 5. An urn is selected with probability of selecting urn U_k being proportional to (k + 2). A ball is chosen randomly from the selected urn. Then, the probability that the urn $U_{5}$ was selected, given that the ball drawn is white, is equal to

In an examination question paper, all questions are 'True' or 'False' type. These are arranged in such a way that three-fourth of times a question with answer 'True' is followed by a question with answer 'True'. Also two-third of times a question with answer 'False' is followed by a question with answer 'False'. If the question paper has 100 questions, the approximate probability that the correct answer of the 100-th question is 'True', is

Consider an M/G/1 queuing system with arrival rate $\lambda = 1$ and independent and identically distributed successive service times having probability density function g(x) = xe⁻ˣ if x > 0, and 0 otherwise. Define, for $i = 1, 2, \dots, I_{i} = 1$ if the first transition is from i to i − 1, and $I_{i} = 0$ if the first transition from i is i + 1. Then, $Var(I_{2})$ equals

Suppose that X ~ binomial$(9, \theta), 0.7 < \theta < 1$, and Y ~ Poisson$(\lambda), \lambda > 0$. If 3E(Y) = E(X), then which of the following is true?

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 2)$ be a random sample from a gamma distribution with shape parameter $\alpha > 0$ and scale parameter $\beta = 1$. For a suitable constant C, the rejection region of the most powerful test for testing $H_{0} : \alpha = 1$ against $H_{1} : \alpha = 2$ is of the form

Suppose $\varphi$ is a most powerful test of size 0.05 for testing a simple null hypothesis $H_{0}$ against a simple alternative hypothesis $H_{1}$. If the power of the test is 0.4, then which of the following is true?

Let $X_{1}, X_{2}, \dots, X_{n} (n \ge 2)$ be a random sample from Uniform$(\theta, 1 - \theta)$ distribution, where $-\infty < \theta < 1/2$. The maximum likelihood estimator of $\theta$ is

Let $X_{1}, X_{2}, \dots, X_{5}$ be a random sample of size 5 from an absolutely continuous distribution having median M. Let S denote the number of $X_{i}'s$ greater than 0. For testing $H_{0} : M = 0$ against $H_{1} : M > 0$, let $\varphi(X) = 1$ if $S > c, \nu$ if S = c, and 0 if S < c be a test of size $\alpha = 0.05$, where $\nu \in [0, 1]$ and c ∈ {−1, 0, …, 5} are fixed constants. Then $c + \nu$ equals

Consider a multiple linear regression model $Y_{i} = \beta_{0} + \beta_{1}x_{i1} + \cdots + \beta_p x_ip + \varepsilon_{i}, 1 \le i \le n, n > (p + 1)$, where the errors $\varepsilon_{i}'s$ are uncorrelated with zero mean and finite variance $\sigma^{2} > 0$. Here, $Y_{i}$ is the i-th response. Let Ŷ$_{i}$ be the i-th predicted response by the least squares estimation method, and let $\varepsilon$̂$_{i} = Y_{i} -$ Ŷ$_{i}, 1 \le i \le n$. Then, which of the following statements is true?

Let X, Y and Z be random variables such that S = [[X, Y], [Y, Z]] ~ $W_{2}(10, \sum)$, where $W_{2}$ denotes the Wishart distribution and $\sum = [[1, 1/2], [1/2, 1]]$. Define $T = Z - Y^{2}/X$. Then, Var(T) equals

Let P be the population proportion of units possessing a certain attribute in a population of N units. Let p be the sample proportion in a simple random sample (without replacement) of n units, (2 ≤ n < N). Then an unbiased estimator of P(1 − P) is

Let $A_{1}, A_{2}, A_{3}$ be events satisfying $0 < P(A_{i}) < 1$ for i = 1, 2, 3. Which of the following statements is true?