CSIR NET Maths previous year question papers

Suppose that we have a data set consisting of 2n + 1 observations for some $n \in \mathbb{N}$. Value of each observation is either x or x + r, where $x \in \mathbb{N}, r \ge 0$. Then, which of the following statements is always true?

A biased six-faced die is tossed once. Suppose that the probability of any prime number showing up is twice that of any non-prime number showing up. Then, the probability that an odd number will show up is

Suppose the distribution of X given $\theta$ is normal with mean $\theta$ and variance 15. Further, let the prior (improper) distribution of $\theta$ be proportional to $1, -\infty < \theta < \infty$. If the observed value of X is 13, then which of the following statements is true?

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 be a random sample of size 1 from the probability density function f(x|$\theta) = (3/\theta^{3})(\theta - x)^{2}$ if $0 < x < \theta$, and 0 otherwise. If $(X/(1 - \lambda_{1}), X/(1 - \lambda_{2}))$ is a confidence interval for $\theta$ with confidence coefficient $1 - \alpha$, where $\lambda_{i} \in (0, 1), i = 1, 2, \lambda_{1} < \lambda_{2}$, and $\alpha \in (0, 1)$, then which of the following statements is true?

Let X, Y, and Z be independent Normal random variables with means −1, 0, and 1, respectively, and variances 1, 1, and 3, respectively. Which of the following random variables has a Cauchy distribution with location parameter 0 and scale parameter 1?

Let {$Y_{n} : n \ge 1$} be a sequence of independent and identically distributed random variables, where $Y_{1}$ ~ Bernoulli(1/2). Define $Z = \sum_{n=1}^\infty$ $4Y_{n}/5^{n}$. Then, which of the following statements is true?

Let X be a random variable with the cumulative distribution function F_X(x) = 0 if x < 0, F_X(x) = (x + 2)/5 if 0 ≤ x < 2, and F_X(x) = 1 if x ≥ 2. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be a random sample from $N(\theta, 1), \theta \in \mathbb{R}$. If $\theta$̂ is the Bayes estimator of $\theta$ with respect to some prior $\pi(\theta)$ and loss function $L(\theta, d)$. Then, which of the following statements are true?

Let X and Y be independent Poisson random variables with means 4 and 2, respectively. Then, which of the following statements are true?

A system has two components $C_{1}$ and $C_{2}$ put in parallel. The components $C_{1}$ and $C_{2}$ have independent lifetimes $X_{1}$ and $X_{2}$, respectively. The probability distribution of X_j is exponential with mean 1/j, j = 1, 2. Suppose that R(t) and h(t) are the reliability and the hazard rate functions of the system, respectively. Then, which of the following statements are true?

Let X and Y be independent and identically distributed N(0, 1) random variables. Let $S = X^{2} + Y^{2}$ and T = $e^{-(X^{2}+Y^{2})/2}$. Then, which of the following statements are true?

Let $X_{1}, X_{2}, \dots, X_{n}$ be independent and identically distributed random variables with the probability density function f(x|$\theta) =$ $e^{-(x-\theta)}$ if $x \ge \theta$, and 0 otherwise, where $\theta \in \mathbb{R}$ is the unknown parameter. Further, let $Y = \min(X_{1}, X_{2}, \dots, X_{n})$. Which of the following are confidence intervals for $\theta$ with the confidence coefficient $(1 - \alpha)$, where $\alpha \in (0, 1)$?

Suppose X∣$\theta$ ~ Binomial$(7,\theta), 0<\theta<1$, and the prior distribution of $\theta$ is Beta$(\alpha,\beta)$ where $\alpha>0$ and $\beta>0$ are known. Then which of the following statements MAY NOT be 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 and Y be two independent random variables such that the moment generating functions of X and Y are $M_X(t)=e^(3(e^t-1)), t\in\mathbb{R}$, and $M_Y(t)=((1/2)e^(-3t)+(1/2)e^(3t))^{2}, t\in\mathbb{R}$, respectively. Then which of the following statements are true?

Suppose two fair dice are thrown independently at random. Let X and Y be the numbers on the upper face of the first die and that of the second die, respectively. Then which of the following statements are true?

Consider a series system comprising four components, having independent and identically distributed lifetimes with hazard rate $\lambda(t)=1/(1+t), t>0$, and survival function S(t), t>0. If Y denotes the lifetime of the series system, then which of the following statements are 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

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

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

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

Let X be a random variable with cumulative distribution function given by F(x) = 0 if x < 0; F(x) = (x + 1)/3 if 0 ≤ x < 1; F(x) = 1 if x ≥ 1. Then the value of P(1/3 < X < 3/4) + P(X = 0) is equal to

Let a point P be chosen at random on the line segment AB of length $\alpha$. Let $Z_{1}$ and $Z_{2}$ denote the lengths of line segments AP and BP respectively. Then the value of E(|$Z_{1} - Z_{2}$|) is