List of practice Questions

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

Let $ x_1 = 1.1, x_2 = 2.2, x_3 = 3.3 $ be the observed values of a random sample of size three from a distribution with the probability density function \[ f(x; \theta) = \begin{cases} \dfrac{1}{\theta} e^{-x/\theta}, & x > 0, \\ 0, & \text{otherwise}, \end{cases} \] where $ \theta \in \Theta = \{ 1, 2, \dots \} $ is the unknown parameter. Then the maximum likelihood estimate of $ \theta $ equals .........

Let $ M = \sum_{i=1}^{4} X_i X_i^T $, where \[ X_1^T = [1 \ -1 \ 1 \ 0], X_2^T = [1 \ 1 \ 0 \ 1], X_3^T = [1 \ 3 \ 1 \ 0] \, \text{and} \, X_4^T = [1 \ 1 \ 1 \ 0]. \] Then the rank of $ M $ equals ...............

Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function such that $ f^\prime $ is continuous on $ \mathbb{R} $ with $ f^\prime(3) = 18 $. Define \[ g_n(x) = n \left( f \left( x + \frac{5}{n} \right) - f \left( x - \frac{2}{n} \right) \right). \] Then $ \lim_{n \to \infty} g_n(3) $ equals ...............

Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function with $ \lim_{x \to \infty} f(x) = \infty $ and $\lim_{x \to \infty} f'(x) = 2. $ Then \[ \lim_{x \to \infty} \left( 1 + \frac{f(x)}{x^2} \right)^x \] equals ..............

The value of \[ \int_0^\frac{\pi}{2} \left( \int_0^x e^{\sin y} \sin x \, dy \right) dx \] equals .............

Let $ X $ be a random variable with the probability density function \[ f(x) = \begin{cases} 4x^k, & 0 < x < 1, \\ x - \dfrac{x^2}{2}, & 1 \leq x < 2, \\ 0, & \text{otherwise}, \end{cases} \] where $ k $ is a positive integer. Then $ P\left( \frac{1}{2} \leq X \leq \frac{3}{2} \right) \textbf{ equals ............} $

Let $ X $ and $ Y $ be two discrete random variables with the joint moment generating function \[ M_{X,Y}(t_1, t_2) = \left( \frac{1}{3} e^{t_1} + \frac{2}{3} \right)^2 \left( \frac{2}{3} e^{t_2} + \frac{1}{3} \right)^3, t_1, t_2 \in \mathbb{R}. \] Then $ P(2X + 3Y > 1) $ equals .........

Let $ X_1, X_2, X_3 $ and $ X_4 $ be i.i.d. discrete random variables with the probability mass function

\[ P(X_1 = n) = \begin{cases} \dfrac{3^{n-1}}{4^n}, & n = 1, 2, \dots, \\ 0, & \text{otherwise}. \end{cases} \]

Then $ P(X_1 + X_2 + X_3 + X_4 = 6) $ equals ...............

What is the voltage gain in a common emitter amplifier, where input resistance is $3 \,\Omega$ and load resistance $24\,\Omega \, \beta = 0.6 $?

Infinite number of masses, each $1 \,kg$ are placed along the $x-$ axis at $x=\pm 1 m, \pm 2 m, \pm 4 m, \pm 8 m, \pm 16 \,m, \ldots$ the magnitude of the resultant gravitational potential in terms of gravitational constant $G$ at the origin $(x=0)$ is

If a stone of mass $0.05 \,kg$ is thrown out a window of a train moving at a constant speed of $100\, km/h$ then magnitude of the net force acting on the stone is

In the circuit shown, the symbols have their usual meanings. The cell has emf $E$. $X$ is initially joined to $Y$ for a long time. Then, $X$ is joined to $Z$. The maximum charge on $C$ at any later time will be