List of practice Questions

Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation such that \( T\left( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) and \( T\left( \begin{bmatrix} 3 \\ 2 \end{bmatrix} \right) = \begin{bmatrix} a \\ b \end{bmatrix} \). Then \( \alpha + \beta + a + b \) equals

Let \( X_1, X_2, X_3, X_4 \) be a random sample from a discrete distribution with the probability mass function \[ P(X = 0) = 1 - P(X = 1) = 1 - p, \quad 0<p<1. \] To test the hypothesis \[ H_0 : p = \frac{3}{4} \quad \text{against} \quad H_1 : p = \frac{4}{5}, \] consider the test: Reject \( H_0 \) if \( X_1 + X_2 + X_3 + X_4>3 \). Let the size and power of the test be denoted by \( \alpha \) and \( \gamma \), respectively. Then \( \alpha + \gamma \) (round off to 2 decimal places) equals ________

In an ethnic group, 30% of the adult male population is known to have heart disease. A test indicates high cholesterol level in 80% of adult males with heart disease. But the test also indicates high cholesterol levels in 10% of the adult males with no heart disease. Then the probability (round off to 2 decimal places) that a randomly selected adult male from this population does not have heart disease given that the test indicates high cholesterol level equals .............

Let \( X \) and \( Y \) be jointly distributed continuous random variables, where \( Y \) is positive valued with \( E(Y^2) = 6 \). If the conditional distribution of \( X \) given \( Y = y \) is \( U(1 - y, 1 + y) \), then \( \text{Var}(X) \) equals .................

Let \( X_1, X_2, \dots, X_{10} \) be i.i.d. \( N(0, 1) \) random variables. If \( T = X_1^2 + X_2^2 + \dots + X_{10}^2 \), then \( E\left( \frac{1}{T} \right) \) equals ..................

Let \( X_1, X_2, X_3 \) be a random sample from \( N(\mu_1, \sigma_1^2) \) distribution and \( Y_1, Y_2, Y_3 \) be a random sample from \( N(\mu_2, \sigma_2^2) \) distribution. Also, assume that \( (X_1, X_2, X_3) \) and \( (Y_1, Y_2, Y_3) \) are independent. Let the observed values of \( \sum_{i=1}^{3} \left[ X_i - \frac{1}{3} (X_1 + X_2 + X_3) \right]^2 \) and \( \sum_{i=1}^{3} \left[ Y_i - \frac{1}{3} (Y_1 + Y_2 + Y_3) \right]^2 \) be 1 and 5, respectively. Then the solution for the 90% confidence interval for \( \mu_1 - \mu_2 \) equals ................

Evaluate the limit \[ \lim_{n \to \infty} \left[ n - \frac{n}{e} \left( 1 + \frac{1}{n} \right)^n \right] \text{ equals ............} \]

The volume (round off to 2 decimal places) of the region in the first octant \( (x \geq 0, y \geq 0, z \geq 0) \) bounded by the cylinder \( x^2 + y^2 = 4 \) and the planes \( z = 2 \) and \( y + z = 4 \) equals .................

Two fair dice are tossed independently and it is found that one face is odd and the other one is even. Then the probability (round off to 2 decimal places) that the sum is less than 6 equals .............

Let \( X \) be a random variable with the moment generating function \[ M_X(t) = \left( \frac{e^{t/2} + e^{-t/2}}{2} \right)^2, \quad -\infty<t<\infty. \] Using Chebyshev's inequality, the upper bound for \( P \left( |X|>\frac{2}{\sqrt{3}} \right) \) equals ...............

In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals ............

Let \( X \) be the number of heads obtained in a sequence of 10 independent tosses of a fair coin. The fair coin is tossed again \( X \) number of times independently, and let \( Y \) be the number of heads obtained in these \( X \) number of tosses. Then \( E(X + 2Y) \) equals ............

Evaluate the following limit (round off to 2 decimal places): \[ \lim_{n \to \infty} \frac{\sqrt{n+1} + \sqrt{n+2} + \cdots + \sqrt{n+n}}{\sqrt{n}} \]

The value (round off to 2 decimal places) of the double integral \[ \int_0^9 \int_{\sqrt{x}}^3 \frac{1}{1 + y^3} \, dy \, dx \] equals .............

Let \( X_1, X_2, \dots, X_n \) be a random sample from a \( U(\theta, 0) \) distribution, where \( \theta<0 \). If \( T_n = \min(X_1, X_2, \dots, X_n) \), then which of the following sequences of estimators is (are) consistent for \( \theta \)?

Let \( P \) be an \( n \times n \) non-null real skew-symmetric matrix, where \( n \) is even. Which of the following statements is (are) always TRUE?

Let \( X_1, X_2, \dots, X_n \) be a random sample from a \( N(\theta, 1) \) distribution. To test \( H_0: \theta = 0 \) against \( H_1: \theta = 1 \), assume that the critical region is given by \[ \frac{1}{n} \sum_{i=1}^n X_i \geq \frac{3}{4}. \] Then the minimum sample size required so that \( P(\text{Type I error}) \leq 0.05 \) is

Let \(\{x_n\}_{n \geq 1}\) be a sequence of positive real numbers such that the series \(\sum_{n=1}^{\infty} x_n\) converges. Which of the following statements is (are) always TRUE?

Let \(f: \mathbb{R} \to \mathbb{R}\) be continuous on \(\mathbb{R}\) and differentiable on \((- \infty, 0) \cup (0, \infty)\). Which of the following statements is (are) always TRUE?

Let -1 and 1 be the observed values of a random sample of size two from \( N(\theta, \theta) \) distribution. The maximum likelihood estimate of \( \theta \) is

Let \( X_1 \) and \( X_2 \) be a random sample from a continuous distribution with the probability density function \[ f(x) = \frac{1}{\theta} e^{-\frac{x - \theta}{\theta}}, \quad x>\theta \] If \( X_{(1) = \min \{ X_1, X_2 \} \) and \( \overline{X} = \frac{X_1 + X_2}{2} \), then which one of the following statements is TRUE?}

Let \( X_1, X_2, X_3 \) be i.i.d. \( U(0,1) \) random variables. Then \[ P(X_1>X_2 + X_3) \]

Let \( X \) and \( Y \) be i.i.d. \( U(0,1) \) random variables. Then \( E(X|X>Y) \) equals

The lifetime (in years) of bulbs is distributed as an \( \text{Exp}(1) \) random variable. Using Poisson approximation to the binomial distribution, the probability (rounded off to 2 decimal places) that out of the fifty randomly chosen bulbs at most one fails within one month equals

Let \( X \) follow a beta distribution with parameters \( m (> 0) \) and 2. If \( P(X \leq \frac{1}{2}) = \frac{1}{2} \), then \( \text{Var}(X) \) equals