List of top Questions asked in IIT JAM MS

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

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