List of top Statistics Questions asked in Indian Institute Of Technology Joint Admission Test for MSc

A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals

Let the random vector \( (X, Y) \) have the joint probability density function
\[f(x, y) = \begin{cases} \frac{1}{x}, & \text{if } 0 < y < x < 1 \\0, & \text{otherwise} \end{cases}.\]
Then \( \text{Cov}(X, Y) \) is equal to:

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 \( n \in \mathbb{N} \), let \( X_n \) be a random variable having the \( \text{Bin}(n, \frac{1}{4}) \) distribution. Then
\[\lim_{n \to \infty} \left[ P(X_n \leq 2n - \frac{\sqrt{3n}}{8}) + P\left(\frac{n}{6} \leq X_n \leq \frac{n}{3}\right) \right]\]
is equal to (You may use \( \Phi(0.5) = 0.6915 \), \( \Phi(1) = 0.8413 \), \( \Phi(1.5) = 0.9332 \), \( \Phi(2) = 0.9772 \)).

Let \( f_1(x) \) be the probability density function of the \( N(0,1) \) distribution, and \( f_2(x) \) be the probability density function of the \( N(0, 6) \) distribution. Let \( Y \) be a random variable with probability density function
\[f(x) = 0.6 f_1(x) + 0.4 f_2(x), \quad -\infty < x < \infty.\]
Then \( \text{Var}(Y) \) is equal to:

Let \( X_1, X_2, X_3 \) be a random sample from a Poisson distribution with mean \( \lambda \), \( \lambda > 0 \). For testing \( H_0: \lambda = \frac{1}{8} \) against \( H_1: \lambda = 1 \), a test rejects \( H_0 \) if and only if \( X_1 + X_2 + X_3 > 1 \). The power of this test is equal to __________ (round off to 2 decimal places).

For \( n \in \mathbb{N} \), let \( Z_n \) be the smallest order statistic based on a random sample of size \( n \) from the \( U(0,1) \) distribution. Let \( nZ_n \xrightarrow{d} Z \) as \( n \to \infty \) for some random variable \( Z \). Then \( P(Z \leq \ln 3) \) is equal to:

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).

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?

Suppose that the random variable \( X \) has an \( \text{Exp}\left(\frac{1}{5}\right) \) distribution, and for any \( x > 0 \), the conditional distribution of the random variable \( Y \), given \( X = x \), is \( N(x, 2) \). Then \( \text{Var}(X + Y) \) is equal to:

Let \(X_1, X_2, \ldots, X_{10}\) be a random sample from the \(N(3,4)\) distribution, and let \(Y_1, Y_2, \ldots, Y_{15}\) be a random sample from the \(N(-3,6)\) distribution. Assume that the two samples are drawn independently. Define:
\[ \bar{X} = \frac{1}{10} \sum_{i=1}^{10} X_i \]
\[ \bar{Y} = \frac{1}{15} \sum_{j=1}^{15} Y_j \]
\[ S = \sqrt{\frac{1}{9} \sum_{i=1}^{10} (X_i - \bar{X})^2} \]
Then the distribution of \( U = \frac{\sqrt{5}(\bar{X} + \bar{Y})}{S} \) is:

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).

For \( n \geq 2 \), let \( \epsilon_1, \epsilon_2, \ldots, \epsilon_n \) be i.i.d. random variables having the \( N(0,1) \) distribution. Consider \( n \) independent random variables \( Y_1, Y_2, \ldots, Y_n \) defined by \( Y_i = \beta + \epsilon_i \), \( i = 1,2, \ldots, n \), where \( \beta \in \mathbb{R} \). Define
\[ \bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i \]
\[ T_1 = \frac{2\bar{Y}}{n+1} \]
\[ T_2 = \frac{1}{n} \sum_{i=1}^{n} \frac{Y_i}{i} \]
Then which of the following statements is NOT correct?

Let \( X \) be a random variable having the Poisson distribution with mean \( 1 \). Let \( g: \mathbb{N} \cup \{0\} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} 1 & \text{if } x \in \{0, 2\} \\ 0 & \text{if } x \notin \{0, 2\} \end{cases}\]
Then \( E(g(X)) \) is equal to:

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?

Let \( Y \) be a continuous random variable such that \( P(Y > 0) = 1 \) and \( \mathbb{E}(Y) = 1 \). For \( p \in (0, 1) \), let \( \xi_p \) denote the \( p \)th quantile of the probability distribution of the random variable \( Y \). Then which of the following statements is always correct?

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).

Two fair coins \( S_1 \) and \( S_2 \) are tossed independently once. Let the events \( E, F, \) and \( G \) be defined as follows:
\( E \): Head appears on \( S_1 \)
\( F \): Head appears on \( S_2 \)  
\( G \): The same outcome (head or tail) appears on both \( S_1 \) and \( S_2 \)  
Then which of the following statements is NOT correct?

Let \( X \) and \( Y \) be continuous random variables having the joint probability density function
\[ f(x, y) = \begin{cases} e^{-x}, & \text{if } 0 \leq y < x < \infty \\0, & \text{otherwise}\end{cases} \]
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 \( X \) be a random variable such that \( X \) and \( -X \) have the same distribution. Let \( Y = X^2 \) be a continuous random variable with the probability density function
\[g(y) = \begin{cases} \frac{e^{-y/2}}{\sqrt{2\pi y}}, & \text{if } y > 0 \\0, & \text{if } y \leq 0 \end{cases}.\]
Then \( E((X - 1)^4) \) is equal to:

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).

For \( n \geq 2 \), let \( X_1, X_2, \ldots, X_n \) be a random sample from a distribution with \( E(X_1) = 0 \), \( \text{Var}(X_1) = 1 \), and \( E(X_1^4) < \infty \). Let \[ \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i \] and \[ S_n^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X}_n)^2. \]
Then which of the following statements is/are always correct?

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 continuous random variable having the 𝑈(−2, 3) distribution. Then which of the following statements is correct?