List of top Statistics Questions on Statistics asked in IIT JAM MS

The following data represent the lifetimes (in months) of a sample of eleven bulbs:

Two different brands, namely Brand A and Brand B, of mobile batteries are tested and the following data on their lifetimes in months are obtained:

Consider the problem of testing \( H_0: X \sim f_0 \) against \( H_1: X \sim f_1 \) based on a sample of size 1, where \[ f_0(x) = \begin{cases} 1, & 0 \le x \le 1, \\ 0, & \text{otherwise,} \end{cases} \quad f_1(x) = \begin{cases} 2x, & 0 \le x \le 1, \\ 0, & \text{otherwise.} \end{cases} \] Then, the probability of Type II error of the most powerful test of size \(\alpha = 0.1\) is equal to

Let \( X \) be a continuous random variable with CDF \[ F(x) = \begin{cases} 0, & x < 0 \\ a x^2, & 0 \le x < 2 \\ 1, & x \ge 2 \end{cases} \] for some real constant \( a \). Then \( E(X) \) is equal to:

Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be a function defined by \[ f(x,y) = \begin{cases} \dfrac{y^3}{x^2 + y^2}, & (x,y) \ne (0,0) \\ 0, & (x,y) = (0,0) \end{cases} \] Let \( f_x(x,y) \) and \( f_y(x,y) \) denote the first-order partial derivatives of \( f(x,y) \) with respect to \( x \) and \( y \) respectively. Then, which of the following statements is FALSE?

The number of real roots of the polynomial \[ f(x) = x^{11} - 13x + 5 \] is ..................

Let \( X \) be a random variable with moment generating function \[ M_X(t) = \frac{1}{12} + \frac{1}{6} e^t + \frac{1}{3} e^{2t} + \frac{1}{4} e^{-t} + \frac{1}{6} e^{-2t}, \quad t \in \mathbb{R}. \] Then, \( 8E(X) \) is equal to ...............

A sample of size \(n\) is drawn randomly (without replacement) from an urn containing \(5n^2\) balls, of which \(2n^2\) are red balls and \(3n^2\) are black balls. Let \(X_n\) denote the number of red balls in the selected sample. If \(\ell = \lim_{n \to \infty} \frac{E(X_n)}{n}\) and \(m = \lim_{n \to \infty} \frac{\text{Var}(X_n)}{n}\), then which of the following statements is/are TRUE?

Let \( X_1, X_2, ..., X_n \) be a random sample from \( N(\theta, 1) \), where \( \theta \in (-\infty, \infty) \) is unknown. Consider the problem of testing \( H_0: \theta \le 0 \) against \( H_1: \theta>0 \). Let \( \beta(\theta) \) denote the power function of the likelihood ratio test of size \( \alpha \) (\(0<\alpha<1\)) for testing \( H_0 \) against \( H_1 \). Then, which of the following statements is/are TRUE?

Let \( X \) and \( Y \) be random variables having chi-square distributions with 6 and 3 degrees of freedom respectively. Then, which of the following statements is TRUE?

Let \(X\) be a random variable with \[ f(x) = \frac{1}{2} e^{-|x|}, \quad -\infty<x<\infty \] Then, which of the following statements is FALSE?

Let \(X\) and \(Y\) be independent \(N(0,1)\) random variables and \(Z = \left|\frac{X}{Y}\right|\). Then, which of the following expectations is finite?

Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) = x^7 + 5x^3 + 11x + 15 \). Then, which of the following statements is TRUE?

Let \( X \) be a random variable with the cumulative distribution function

\[ F(x) = \begin{cases} 0, & x < 0 \\ 1 + x^2, & 0 \leq x < 1 \\ \frac{10}{3} + x^2, & 1 \leq x < 2 \\ 1, & x \geq 2 \end{cases} \]

Which of the following statements is (are) TRUE?

Let \( X \) and \( Y \) be i.i.d. \( \text{Exp}(\lambda) \) random variables. If \( Z = \max\{X - Y, 0\} \), then which of the following statements is (are) TRUE?

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 ................

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 .................

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 ............

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 \( X_1, X_2, \dots, X_n \) 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 -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 \( X \) and \( Y \) be i.i.d. \( U(0,1) \) random variables. Then \( E(X|X>Y) \) 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