List of top Statistics Questions

Let \( X_1, X_2, \dots, X_n \ (n \ge 2) \) be a random sample from a distribution with probability density function \[ f(x; \theta) = \begin{cases} \theta x^{\theta - 1}, & 0 \le x \le 1, \\ 0, & \text{otherwise,} \end{cases} \] where \( \theta \in (0, \infty) \) is unknown. Then, which of the following statements is/are TRUE?

Let \( f_0 \) and \( f_1 \) be the probability mass functions given by: \[ \begin{array}{c|cccccc} x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline f_0(x) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.5 \\ f_1(x) & 0.1 & 0.1 & 0.2 & 0.2 & 0.2 & 0.2 \end{array} \] Consider the problem of testing the null hypothesis \(H_0: X \sim f_0\) against \(H_1: X \sim f_1\) based on a single sample \(X\). If \( \alpha \) and \( \beta \), respectively, denote the size and power of the test with critical region \( \{x \in \mathbb{R} : x > 3\} \), then \(10(\alpha + \beta)\) is equal to .........................

Let \( 5, 10, 4, 15, 6 \) be an observed random sample of size 5 from a distribution with probability density function \[ f(x; \theta) = \begin{cases} e^{-(x - \theta)}, & x \ge \theta \\ 0, & \text{otherwise} \end{cases} \] where \( \theta \in (-\infty, 3] \) is unknown. Then, the maximum likelihood estimate (MLE) of \( \theta \) based on the observed sample is equal to ..............

Let \( X_1 \) and \( X_2 \) be independent \( N(0,1) \) random variables. Define \[ \text{sgn}(u) = \begin{cases} -1, & \text{if } u < 0 \\ 0, & \text{if } u = 0 \\ 1, & \text{if } u > 0 \end{cases} \] Let \( Y_1 = X_1 \, \text{sgn}(X_2) \) and \( Y_2 = X_2 \, \text{sgn}(X_1) \). If the correlation coefficient between \(Y_1\) and \(Y_2\) is \(\alpha\), then \(\pi \alpha\) is equal to ............

Let the random vector \((X, Y)\) have the joint probability mass function \[ f(x, y) = \begin{cases} \binom{10}{x} \binom{5}{y} \left(\frac{1}{4}\right)^{x - y + 5} \left(\frac{3}{4}\right)^{y - x + 10}, & x = 0, 1, \ldots, 10; \; y = 0, 1, \ldots, 5 \\ 0, & \text{otherwise} \end{cases} \] Let \( Z = Y - X + 10 \). If \( \alpha = E(Z) \) and \( \beta = \text{Var}(Z) \), then \( 8\alpha + 48\beta \) 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 \( X \) be a random variable with pdf \[ f(x) = \begin{cases} e^{-x}, & x > 0 \\ 0, & x \le 0 \end{cases} \] Define \( Y = \lfloor X \rfloor \), the greatest integer less than or equal to \( X \). Then \(E(Y^2)\) 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?

Let \(\{X_n\}_{n \ge 1}\) be i.i.d. random variables with \[ f(x) = \begin{cases} 1, & 0<x<1 \\ 0, & \text{otherwise} \end{cases} \] Then, the value of the limit \[ \lim_{n \to \infty} P\left( -\frac{1}{n}\sum_{i=1}^{n} \ln X_i \le 1 + \frac{1}{\sqrt{n}} \right) \] is equal to:

Let \[ S = \{ (x, y) \in \mathbb{R}^2 : 0 \le x \le \pi, \min(\sin x, \cos x) \le y \le \max(\sin x, \cos x) \}. \] If \(\alpha\) is the area of \(S\), then the value of \(2\sqrt{2}\,\alpha\) is equal to ............

Let \(\phi : (-1, 1) \to \mathbb{R}\) be defined by \[ \phi(x) = \int_{x^7}^{x^4} \frac{1}{1 + t^3} \, dt. \] If \[ \alpha = \lim_{x \to 0} \frac{\phi(x)}{e^{x^4} - 1}, \] then \(42\alpha\) is equal to ...............

Let \[ \alpha = \lim_{n \to \infty} \left(1 + n \sin \frac{3}{n^2}\right)^{2n}. \] Then, \(\ln \alpha\) is equal to ................

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

Let \( S \subseteq \mathbb{R}^2 \) be the region bounded by the parallelogram with vertices at the points \( (1,0), (3,2), (3,5) \) and \( (1,3) \). Then, the value of the integral \[ \iint_S (x + 2y) \, dx \, dy \] is equal to ..............

Let \( E_1, E_2, E_3 \) and \( E_4 \) be four independent events such that \[ P(E_1) = \frac{1}{2}, \quad P(E_2) = \frac{1}{3}, \quad P(E_3) = \frac{1}{4}, \quad P(E_4) = \frac{1}{5}. \] Let \( p \) be the probability that at most two events among \( E_1, E_2, E_3, E_4 \) occur. Then, \( 240p \) is equal to ............

Let \[ a_n = \sum_{k=2}^{n} \binom{n}{k} \frac{2^k (n - 2)^{n - k}}{n^n}, \quad n = 2, 3, \ldots \] Then, \[ e^2 \lim_{n \to \infty} (1 - a_n) \] is equal to ...............

Let \[ A = \left\{ (x, y) \in \mathbb{R}^2 : x^2 - \frac{1}{2\sqrt{\pi}}<y<x^2 + \frac{1}{2\sqrt{\pi}} \right\} \] and let the joint probability density function of \((X, Y)\) be \[ f(x, y) = \begin{cases} e^{-(x - 1)^2}, & (x, y) \in A,
0, & \text{otherwise.} \end{cases} \] Then, the covariance between the random variables \(X\) and \(Y\) is equal to .............

Let \( B \) denote the length of the curve \( y = \ln(\sec x) \) from \( x = 0 \) to \( x = \frac{\pi}{4} \). Then, the value of \( 3\sqrt{2}(e^B - 1) \) is equal to .............

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

Let \( S = \{(x,y) \in \mathbb{R}^2 : 2 \le x \le y \le 4\} \). Then, the value of the integral \[ \iint_S \frac{1}{4 - x} \, dx \, dy \] is ..........

Let \( A = \{(x, y, z) \in \mathbb{R}^3 : 0 \le x \le y \le z \le 1 \} \). Let \( \alpha \) be the value of the integral \[ \iiint_A x y z \, dx \, dy \, dz. \] Then, \( 384 \alpha \) is equal to ..........

Let \( X \) be a random variable having the probability density function \[ f(x) = \frac{1}{8\sqrt{2\pi}} \left( 2 e^{-x^2/2} + 3 e^{-x^2/8} \right), \quad x \in \mathbb{R}. \] Then, \( 4E(X^4) \) is equal to .................

Let \[ \alpha = \lim_{n \to \infty} \sum_{m = n^2}^{2n^2} \frac{1}{\sqrt{5n^4 + n^3 + m}}. \] Then, \( 10\sqrt{5} \, \alpha \) is equal to ...............

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function. Then, which of the following statements is/are necessarily TRUE?

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?