List of practice Questions

Let \( \Theta \) be a random variable having \( U(0, 2\pi) \) distribution. Let \( X = \cos \Theta \) and \( Y = \sin \Theta \). Let \( \rho \) be the correlation coefficient between \( X \) and \( Y \). Then \( 100\rho \) is equal to __________ (answer in integer).

Consider a coin for which the probability of obtaining head in a single toss is \( \frac{1}{3} \). Sunita tosses the coin once. If head appears, she receives a random amount of \( X \) rupees, where \( X \) has the \( \text{Exp}\left( \frac{1}{9} \right) \) distribution. If tail appears, she loses a random amount of \( Y \) rupees, where \( Y \) has the \( \text{Exp}\left( \frac{1}{3} \right) \) distribution. Her expected gain (in rupees) is equal to __________ (answer in integer).

Let \( X \) be a discrete random variable with \( P(X \in \{-5, -3, 0, 3, 5\}) = 1 \). Suppose that \( P(X = -3) = P(X = -5) \), \( P(X = 3) = P(X = 5) \) and \( P(X > 0) = P(X = 0) = P(X < 0) \). Then the value of \( 12P(X = 3) \) is equal to __________ (answer in integer).

Two factories \( F_1 \) and \( F_2 \) produce cricket bats that are labelled. Any randomly chosen bat produced by factory \( F_1 \) is defective with probability 0.5 and any randomly chosen bat produced by factory \( F_2 \) is defective with probability 0.1. One of the factories is chosen at random, and two bats are randomly purchased from the chosen factory. Let the labels on these purchased bats be \( B_1 \) and \( B_2 \). If \( B_1 \) is found to be defective, then the conditional probability that \( B_2 \) is also defective is equal to __________ (round off to 2 decimal places).

For some \( a \leq 0 \) and \( b \in \mathbb{R} \), let
\[A = \begin{pmatrix} 0 & a & b \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \end{pmatrix}.\]
If \( A \) is an orthogonal matrix, then the value of \( a\sqrt{6} + 4b\sqrt{3} \) is equal to __________ (answer in integer).

The value of the integral
\[\int_{0}^{\sqrt{8}} \int_{x^2}^{\sqrt{8 - x^2}} \frac{3\sqrt{x^2 + y^2}}{\sqrt{8}} \, dy \, dx \]
is equal to __________ (answer in integer).

The value of
\[\lim_{n \to \infty} n \left( \sin \frac{1}{2n} - \frac{1}{2} e^{-\frac{1}{n}} + \frac{1}{2} \right) \]
is equal to __________ (answer in integer).

Suppose that the lifetimes (in months) of bulbs manufactured by a company have an \( \text{Exp}(\lambda) \) distribution, where \( \lambda > 0 \). A random sample of size 10 taken from the bulbs manufactured by the company yields the sample mean lifetime \( \bar{x} = 3.52 \) months. Then the uniformly minimum variance unbiased estimate of \( \frac{1}{\lambda} \) based on this sample is equal to __________ months (round off to 2 decimal places).

Let \( X_1, X_2, X_3 \) be i.i.d. random variables from a continuous distribution having probability density function
\[f(x) = \begin{cases} \frac{1}{2}x^{-3}, & \text{if } x > \frac{1}{2} \\0, & \text{if } x \leq \frac{1}{2} \end{cases}.\]
Let \( X_{(1)} = \min\{X_1, X_2, X_3\} \). Then the value of \( 10E(X_{(1)}) \) is equal to __________ (answer in integer).

Let \( X_1, X_2, \ldots, X_{10} \) be a random sample from the \( \text{Exp}(1) \) distribution. Define
\[W = \max\{ e^{-X_1}, e^{-X_2}, \ldots, e^{-X_{10}} \}.\]
Then the value of \( 22E(W) \) is equal to __________ (answer in integer).

For \( n \in \mathbb{N} \), let \( X_1, X_2, \ldots, X_n \) be a random sample from the \( F_{20,40} \) distribution. Then, as \( n \to \infty \), \( \frac{1}{n} \sum_{i=1}^n \frac{1}{X_i} \) converges in probability to __________ (round off to 2 decimal places).

For \( n \in \mathbb{N} \), let \( X_1, X_2, \ldots, X_n \) be a random sample from the Cauchy distribution having probability density function
\[f(x) = \frac{1}{\pi(1 + x^2)}, \quad -\infty < x < \infty.\]
Let \( g: \mathbb{R} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} x, & \text{if } -1000 \leq x \leq 1000 \\0, & \text{otherwise}\end{cases}.\]
Let
\[\alpha = \lim_{n \to \infty} P\left( \frac{1}{n^{3/4}} \sum_{i=1}^n g(X_i) > \frac{1}{2} \right).\]
Then \( 100\alpha \) is equal to __________ (answer in integer).

Let \( X \) be a random variable with the moment generating function \[ M(t) = \frac{(1 + 3e^t)^2}{16}, \quad -\infty < t < \infty. \]
Let \( \alpha = E(X) - \text{Var}(X) \). Then the value of \( 8\alpha \) is equal to __________ (answer in integer).

If 12 fair dice are independently rolled, then the probability of obtaining at least two sixes is equal to __________ (round off to 2 decimal places)

Let \( V = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 \mid x_1 = x_2 \} \). Consider \( V \) as a subspace of \(\mathbb{R}^4\) over the real field. Then the dimension of \( V \) is equal to __________ (answer in integer).

Let \( f(x) = \int_{2-2x}^{x^2-1} e^{t^2-t} \, dt \) for all \( x \in \mathbb{R} \). If \( f \) is decreasing on \( (0, m) \) and increasing on \( (m, \infty) \), then the value of \( m \) is equal to __________ (answer in integer).

The radius of convergence of the power series \( \sum_{n=0}^{\infty} \frac{2^n (x+3)^n}{5^n} \) is equal to __________ (answer in integer).

Let \(\theta_0\) and \(\theta_1\) be real constants such that \(\theta_1 > \theta_0\). Suppose that a random sample is taken from a \(N(\theta, 1)\) distribution, \(\theta \in \mathbb{R}\). For testing \(H_0: \theta = \theta_0\) against \(H_1: \theta = \theta_1\) at level 0.05, let \(\alpha\) and \(\beta\) denote the size and the power, respectively, of the most powerful test, \(\psi_0\). Then which of the following statements is/are correct?

Let \( X_1, X_2, \ldots, X_{50} \) be a random sample from a \( N(0, \sigma^2) \) distribution, where \( \sigma > 0 \). Define
\[ \bar{X}_e = \frac{1}{25} \sum_{i=1}^{25} X_{2i}, \] \[ \bar{X}_o = \frac{1}{25} \sum_{i=1}^{25} X_{2i-1}, \] \[ S_e = \sqrt{\frac{1}{24} \sum_{i=1}^{25} (X_{2i} - \bar{X}_e)^2}, \] and \[ S_o = \sqrt{\frac{1}{24} \sum_{i=1}^{25} (X_{2i-1} - \bar{X}_o)^2}. \]
Then which of the following statements is/are correct?

Let \( X \) and \( Y \) be independent random variables having \( \text{Bin}(18, 0.5) \) and \( \text{Bin}(20, 0.5) \) distributions, respectively. Further, let \( U = \min\{X, Y\} \) and \( V = \max\{X, Y\} \). Then which of the following statements is/are correct?

The design of a 1200 capacity concert hall considers 1/3rd female audience and 2/3rd male audience. The Table below shows the guideline for calculating Water Closet requirements.

Fixture	Male	Female
Water closet	1 number per 100 up to 400 and over 400, 1 number for every 250 or part thereof	3 numbers per 100 up to 200 population and over 200, 2 numbers for every 100 or part thereof

Using the above guideline, the number of Water Closets required for the total audience is _______(in integer).

Let \( X \) be a continuous random variable with a probability density function \( f \) and the moment generating function \( M(t) \). Suppose that \( f(x) = f(-x) \) for all \( x \in \mathbb{R} \) and the moment generating function \( M(t) \) exists for \( t \in (-1, 1) \). Then which of the following statements is/are correct?

Consider four dice \( D_1, D_2, D_3, \) and \( D_4 \), each having six faces marked as follows:

Die	Marks on faces
\(D_1\)	4, 4, 4, 4, 0, 0
\(D_2\)	3, 3, 3, 3, 3, 3
\(D_3\)	6, 6, 2, 2, 2, 2
\(D_4\)	5, 5, 5, 1, 1, 1

In each roll of a die, each of its six faces is equally likely to occur. Suppose that each of these four dice is rolled once, and the marks on their upper faces are noted. Let the four rolls be independent. If \( X_i \) denotes the mark on the upper face of the die \( D_i \), \( i = 1, 2, 3, 4 \), then which of the following statements is/are correct?

Let \( A \) be a \( 3 \times 3 \) real matrix. Suppose that 1 and 2 are characteristic roots of \( A \), and 12 is a characteristic root of \( A + A^2 \). Then which of the following statements is/are correct?

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( f(0) = 0 \), \( f(2) = 4 \), \( f(4) = 4 \), and \( f(8) = 12 \). Then which of the following statements is/are correct?