List of practice Questions

Let \( X \) and \( Y \) have the joint probability density function \[ f(x, y) = \begin{cases} 2, & 0 \leq x \leq y \leq 1, \\ 0, & \text{otherwise}. \end{cases} \] Let \( a = E(Y|X = \frac{1}{2}) \) and \( b = \text{Var}(Y|X = \frac{1}{2}) \). Then \( (a, b) \) is 
 

Let \( X \) and \( Y \) have the joint probability mass function \[ P(X = m, Y = n) = \begin{cases} \frac{m + n}{21}, & m = 1, 2, 3; n = 1, 2, \\ 0, & \text{otherwise}. \end{cases}\] Then \( P(X = 2 | Y = 2) \) equals 
 

Let \( X \) and \( Y \) be two independent standard normal random variables. Then the probability density function of \( Z = \frac{|X|}{|Y|} \) is

Let \( X \) and \( Y \) have the joint probability density function \[ f(x, y) = \begin{cases} e^{-y}, & 0 < x < y < \infty, \\ 0, & \text{otherwise}. \end{cases} \] Then the correlation coefficient between \( X \) and \( Y \) equals 
 

Let \( X \) be a random sample from a discrete distribution with the probability mass function \[ f(x; \theta) = P(X = x) = \begin{cases} \frac{1}{\theta}, & x = 1, 2, \dots, \theta, \\ 0, & \text{otherwise}, \end{cases} \] where \( \theta \in \{20, 40\} \) is the unknown parameter. Consider testing \[ H_0: \theta = 40 \,\, \text{against} \,\, H_1: \theta = 20 \] at a level of significance \( \alpha = 0.1 \). Then the uniformly most powerful test rejects \( H_0 \) if and only if 
 

Let \( X_1 \) and \( X_2 \) be a random sample of size 2 from a discrete distribution with the probability mass function \[ f(x; \theta) = P(X = x) = \begin{cases} \theta, & x = 0, \\ 1 - \theta, & x = 1, \end{cases} \] where \( \theta \in \{0.2, 0.4\} \) is the unknown parameter. For testing \( H_0: \theta = 0.2 \,\,\text{against}\,\, H_1: \theta = 0.4, \) consider a test with the critical region \[ C = \{(x_1, x_2) \in \{0, 1\} \times \{0, 1\} : x_1 + x_2 < 2\}. \] Let \( \alpha \) and \( \beta \) denote the probability of Type I error and power of the test, respectively. Then \( (\alpha, \beta) \) is 
 

Let \( \{a_n\}_{n \geq 1} \) be a sequence of real numbers such that \[ a_n = \sum_{k=n+1}^{2n} \frac{1}{k}, n \geq 1. \] Then which of the following statement(s) is (are) true?
Let \( \sum_{n \geq 1} a_n \) be a convergent series of positive real numbers. Then which of the following statement(s) is (are) true?
Let \( \{a_n\}_{n \geq 1} \) be a sequence of real numbers such that \( a_1 = 3 \) and, for \( n \geq 1 \), \[ a_{n+1} = \frac{a_n^2 - 2a_n + 4}{2}. \] Then which of the following statement(s) is (are) true?

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x) = \begin{cases} x^4(2 + \sin \frac{1}{x}), & x \neq 0, \\ 0, & x = 0. \end{cases} \] Then which of the following statement(s) is (are) true? 
 

Let \( P \) be a probability function that assigns the same weight to each of the points of the sample space \( \Omega = \{1, 2, 3, 4\} \). Consider the events \( E = \{1, 2\}, F = \{1, 3\}, G = \{3, 4\}. \) Then which of the following statement(s) is (are) true? 
 

Let \( X_1, X_2, \dots, X_n \), where \( n \geq 5 \), be a random sample from a distribution with the probability density function \[ f(x; \theta) = \begin{cases} e^{-(x - \theta)}, & x \geq \theta, \\ 0, & \text{otherwise}, \end{cases} \] where \( \theta \in \mathbb{R} \) is the unknown parameter. Then which of the following statement(s) is (are) true? 
 

Let \( X_1, X_2, \dots, X_n \) be a random sample from \( U(0, \theta) \), where \( \theta > 0 \) is the unknown parameter. Let \( X_{(n)} = \max{\{X_1, X_2, \dots, X_n\}}. \) Then which of the following is (are) consistent estimator(s) of \( \theta^3 \)? 
 

Let \( X_1, X_2, \dots, X_n \) be a random sample from a distribution with the probability density function \[ f(x; \theta) = \begin{cases} c(\theta) e^{-(x - \theta)}, & x \geq 2\theta, \\ 0, & \text{otherwise}, \end{cases} \] where \( \theta \in \mathbb{R} \) is the unknown parameter. Then which of the following statement(s) is (are) true? 
 

Let \( X_1, X_2, \dots, X_n \) be a random sample from a distribution with the probability density function \[ f(x; \theta) = \begin{cases} \theta^2 x e^{-\theta x}, & x > 0, \\ 0, & \text{otherwise}, \end{cases} \] where \( \theta > 0 \) is the unknown parameter. If \( Y = \sum_{i=1}^n X_i \), then which of the following statement(s) is (are) true? 
 

Let \( X_1, X_2, \dots, X_n \) be a random sample from \( U(\theta, \theta + 1) \), where \( \theta \in \mathbb{R} \) is the unknown parameter. Let \[ U = \max{\{X_1, X_2, \dots, X_n\}} \, \text{and} \, V = \min{\{X_1, X_2, \dots, X_n\}}. \] Then which of the following statement(s) is (are) true? 
 

Let \( \{a_n\}_{n \geq 1} \) be a sequence of real numbers such that \[ a_n = \frac{1 + 3 + 5 + \dots + (2n - 1)}{n!}, n \geq 1. \] Then \( \sum_{n \geq 1} a_n \) converges to ................
Let \[ S = \{(x, y) \in \mathbb{R}^2 : x \geq 0, \sqrt{4 - (x - 2)^2} \leq y \leq \sqrt{9 - (x - 3)^2} \}. \] Then the area of \( S \) equals .............
Let \[ S = \{(x, y) \in \mathbb{R}^2 : |x| + |y| \leq 1\}. \] Then the area of \( S \) equals ...............
Let \[ J = \frac{1}{\pi} \int_0^1 t^{-\frac{1}{2}} (1 - t)^{\frac{3}{2}} \, dt. \] Then the value of \( J \) equals ..............
A fair die is rolled three times independently. Given that 6 appeared at least once, the conditional probability that 6 appeared exactly twice equals ..............

Let \( X \) and \( Y \) be two positive integer-valued random variables with the joint probability mass function \[ P(X = m, Y = n) = \begin{cases} g(m) h(n), & m, n \geq 1, \\ 0, & \text{otherwise}. \end{cases}\] where \( g(m) = \left( \frac{1}{2} \right)^{m-1} \), \( m \geq 1 \), and \( h(n) = \left( \frac{1}{3} \right)^n \), \( n \geq 1 \). Then \( E(XY) \) equals .......... 
 

Let \( E, F \) and \( G \) be three events such that \[ P(E \cap F \cap G) = 0.1, P(G \mid F) = 0.3 \, \text{and} \, P(E \mid F \cap G) = P(E \mid F). \] Then \( P(G \mid E \cap F) \) equals ................ 
 

Let \( A_1, A_2 \) and \( A_3 \) be three events such that \[ P(A_i) = \frac{1}{3}, i = 1, 2, 3; P(A_i \cap A_j) = \frac{1}{6}, 1 \leq i \neq j \leq 3 \text{and} P(A_1 \cap A_2 \cap A_3) = \frac{1}{6}. \] Then the probability that none of the events \( A_1, A_2, A_3 \) occur equals ...........

Let \( X_1, X_2, \dots, X_n \) be a random sample from the distribution with the probability density function \[ f(x) = \frac{1}{4} e^{-|x-4|} + \frac{1}{4} e^{-|x-6|}, x \in \mathbb{R}. \] Then \( \frac{1}{n} \sum_{i=1}^{n} X_i \) converges in probability to ................