List of practice Questions

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

Let \( X \) be a continuous random variable with the probability density function \[ f(x) = \frac{1}{3} x^2 e^{-x^2}, \quad x>0 \] Then the distribution of the random variable \[ W = 2X^2 \quad \text{is} \]

Let \( X \) be a continuous random variable with the probability density function \[ f(x) = \frac{e^x}{(1 + e^x)^2}, \quad -\infty<x<\infty \] Then \( E(X) \) and \( P(X>1) \), respectively, are

Let \( E \) and \( F \) be any two independent events with \( 0<P(E)<1 \) and \( 0<P(F)<1 \). Which one of the following statements is NOT TRUE?

Let \( P \) be a \( 3 \times 3 \) non-null real matrix. If there exists a \( 3 \times 2 \) real matrix \( Q \) and a \( 2 \times 3 \) real matrix \( R \) such that \( P = QR \), then

Let \( E, F \), and \( G \) be any three events with \( P(E) = 0.3 \), \( P(F|E) = 0.2 \), \( P(G|E) = 0.1 \). Then \( P(E - (F \cup G)) \) equals

The length of the curve \[ y = \frac{3}{4} x^{4/3} - \frac{3}{8} x^{2/3} + 7 \] from \( x = 1 \) to \( x = 8 \) equals

The volume of the solid generated by revolving the region bounded by the parabola \[ x = 2y^2 + 4 \quad \text{and the line} \quad x = 6 \quad \text{about the line} \quad x = 6 \] is

Evaluate the limit \[ \lim_{n \to \infty} \frac{1 + \frac{1}{2} + \dots + \frac{1}{n}}{(n + e^n)^{1/n} \log_e n} \]

A possible value of \( b \in \mathbb{R} \) for which the equation \[ x^4 + bx^3 + 1 = 0 \] has no real root is