List of top Statistics Questions

Let \( x_1 = 2, x_2 = 1, x_3 = \sqrt{5}, x_4 = \sqrt{2} \) be the observed values of a random sample of size four from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} \frac{1}{2\theta}, & \text{if } -\theta \leq x \leq \theta \\ 0, & \text{otherwise}, \quad \theta > 0 \end{cases} \] Then the method of moments estimate of \( \theta \) is
Let \( \{X_n\}_{n \geq 1} \) be a sequence of i.i.d. random variables with the probability mass function
\[ f(x) = \begin{cases} \frac{1}{4}, & \text{if } x = 4 \\ \frac{3}{4}, & \text{if } x = 8 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i, n = 1, 2, \dots \). If \( \lim_{n \to \infty} P(m \leq \bar{X}_n \leq M) = 1 \), then possible values of \( m \) and \( M \) are
Let \( X_1, X_2, \dots, X_m, Y_1, Y_2, \dots, Y_n \) be i.i.d. \( N(0,1) \) random variables. Then \[ W = \frac{n \sum_{i=1}^m X_i^2}{m \sum_{j=1}^n Y_j^2} \] has
Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x, y) = \begin{cases} c x (1 - x), & 0 < x < y < 1 \\ 0, & \text{otherwise} \end{cases} \] where \( c \) is a positive real constant. Then \( E(X) \) equals
Let \( X_1, X_2, X_3 \) be i.i.d. discrete random variables with the probability mass function
\[ p(k) = \left( \frac{2}{3} \right)^{k-1} \left( \frac{1}{3} \right), \quad k = 1, 2, 3, \dots \] Let \( Y = X_1 + X_2 + X_3 \). Then \( P(Y \geq 5) \) equals
Let the random variable \( X \) have uniform distribution on the interval \( \left( \frac{\pi}{6}, \frac{\pi}{2} \right) \). Then
\[ P(\cos X>\sin X) \] is
Let \( X \) be a continuous random variable with the probability density function
\[ f(x) = \begin{cases} x^3, & 0 < x \leq 1 \\ 3x^5, & x > 1 \\ 0, & \text{otherwise} \end{cases} \] Then \( P\left( \frac{1}{2} \leq X \leq 2 \right) \) equals
Let \( u_n = \frac{(4 - n)}{n} \), \( n \in \mathbb{N} \), and let \( l = \lim_{n \to \infty} u_n \).
Which of the following statements is TRUE?
The imaginary parts of the eigenvalues of the matrix
\[ P = \begin{pmatrix} 3 & 2 & 5 \\ 2 & -3 & 6 \\ 0 & 0 & -3 \end{pmatrix} \] are
If for a suitable \( \alpha>0 \), \[ \lim_{x \to 0} \left( \frac{1}{e^{2x} - 1} - \frac{1}{\alpha x} \right) \] exists and is equal to \( l \) (\( |l|<\infty \)), then \( \alpha = 2, l = -\frac{1}{2} \) is given by
Let \[ P = \int_0^1 \frac{dx}{\sqrt{8 - x^2 - x^3}}. \] Which of the following statements is TRUE?
Let \( X \) be a random variable having a probability density function \( f \in \{ f_0, f_1 \} \), where
\[ f_0(x) = \begin{cases} 1, & 0 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases} \quad \text{and} \quad f_1(x) = \begin{cases} \frac{1}{2}, & 0 \leq x \leq 2 \\ 0, & \text{otherwise} \end{cases} \] For testing the null hypothesis \( H_0 : f = f_0 \) against \( H_1 : f = f_1 \), based on a single observation on \( X \), the power of the most powerful test of size \( \alpha = 0.05 \) equals
If \[ \int_0^1 \int_0^{\sqrt{1 - (y - 1)^2}} f(x, y) \, dx \, dy \] equals \[ \int_0^1 \int_0^x f(x, y) \, dy \, dx, \] then \( \alpha(x) \) and \( \beta(x) \) are
Let \( f: [0, 1] \to \mathbb{R} \) be a function defined as
\[ f(t) = \begin{cases} t^3 \left( 1 + \frac{1}{5} \cos(\log(e^t)) \right), & \text{if } t \in (0,1] \\ 0, & \text{if } t = 0 \end{cases} \] Let \( F: [0, 1] \to \mathbb{R} \) be defined as
\[ F(x) = \int_0^x f(t) \, dt \] Then \( F''(0) \) equals
Consider the function \[ f(x, y) = x^3 - y^3 - 3x^2 + 3y^2 + 7, \, x, y \in \mathbb{R}. \] Then the local minimum (\( m \)) and the local maximum (\( M \)) of \( f \) are given by
Let \( X \) and \( Y \) be continuous random variables with the joint probability density function
\[ f(x, y) = \begin{cases} x + y, & \text{if } 0 < x < 1, 0 < y < 1 \\ 0, & \text{otherwise} \end{cases} \] Then \( P(X + Y > \frac{1}{2}) \) equals
Let \( x_1 = 1.1, x_2 = 0.5, x_3 = 1.4, x_4 = 1.2 \) be the observed values of a random sample of size four from a distribution with the probability density function
\[ f(x|\theta) = \begin{cases} e^{-\theta x}, & \text{if } x \geq \theta \\ 0, & \text{otherwise}, \quad \theta \in (-\infty, \infty) \end{cases} \] Then the maximum likelihood estimate of \( \theta^2 \) is
Let \( X \) be a discrete random variable with the probability mass function
\[ p(x) = k(1 + |x|)^2, \quad x = -2, -1, 0, 1, 2, \] where \( k \) is a real constant. Then \( P(X = 0) \) equals
Let \( \{X_n\}_{n \geq 1} \) be a sequence of i.i.d. random variables having common probability density function
\[ f(x) = \begin{cases} x e^{-x}, & x \geq 0 \\ 0, & \text{otherwise} \end{cases} \] Let \( \bar{X}_n = \frac{1}{n} \sum_{i=1}^{n} X_i \), \( n = 1, 2, \dots \). Then \[ \lim_{n \to \infty} P(\bar{X}_n = 2) \] equals
Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability density function
\[ f(x|\theta) = \frac{1}{\theta} e^{-x/\theta}, \quad x>0, \ \theta>0 \] Which of the following estimators of \( \theta \) has the smallest variance for all \( \theta>0 \)?
Player \( P_1 \) tosses 4 fair coins and player \( P_2 \) tosses a fair die independently of \( P_1 \). The probability that the number of heads observed is more than the number on the upper face of the die, equals
Let \( X_1 \) and \( X_2 \) be i.i.d. continuous random variables with the probability density function
\[ f(x) = \begin{cases} 6x(1 - x), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases} \] Using Chebyshev's inequality, the lower bound of \( P \left( |X_1 + X_2 - 1| \leq \frac{1}{2} \right) \) is
Let \( X \) be a random variable with the moment generating function
\[ M_X(t) = \frac{1}{216} \left( 5 + e^t \right)^3, \quad t \in \mathbb{R}. \] Then \( P(X>1) \) equals
Let \( \{a_n\} \) be a sequence defined as follows:
\[ a_1 = 1 \quad \text{and} \quad a_{n+1} = \frac{7a_n + 11}{21}, \quad n \in \mathbb{N}. \] Which of the following statements is TRUE?
Let \( u, v \in \mathbb{R}^4 \) be such that \( u = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 5 \end{pmatrix} \) and \( v = \begin{pmatrix} 5 \\ 3 \\ 2 \\ 1 \end{pmatrix} \). Then the equation \( u^T x = v \) has