List of top Statistics Questions

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 ..........

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?

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

Let \(\{a_n\}_{n \ge 1}\) be a sequence of real numbers such that \(a_n \ge 1\), for all \(n \ge 1\). Then, which of the following conditions imply the divergence of \(\{a_n\}_{n \ge 1}\)?

Let \(E_1, E_2\) and \(E_3\) be three events such that \(P(E_1) = \frac{4}{5}, P(E_2) = \frac{1}{2}\) and \(P(E_3) = \frac{9}{10}\). Then, which of the following statements is FALSE?

Let \( X_1, X_2, ..., X_n \) be a random sample from \( N(\theta, 1) \), where \( \theta \in (-\infty, \infty) \) is unknown. Consider the problem of testing \( H_0: \theta \le 0 \) against \( H_1: \theta>0 \). Let \( \beta(\theta) \) denote the power function of the likelihood ratio test of size \( \alpha \) (\(0<\alpha<1\)) for testing \( H_0 \) against \( H_1 \). Then, which of the following statements is/are TRUE?

Let \[ S = \sum_{k=1}^{\infty} (-1)^{k-1}\frac{1}{k}\left(\frac{1}{4}\right)^k, \quad T = \sum_{k=1}^{\infty} \frac{1}{k}\left(\frac{1}{5}\right)^k. \] Then, which of the following statements is TRUE?

Let \(E_1, E_2, E_3\) and \(E_4\) be four events such that \[ P(E_i|E_4) = \frac{2}{3}, \; i = 1, 2, 3; \quad P(E_i \cap E_j^c | E_4) = \frac{1}{6}, \; i,j = 1,2,3; \; i \ne j; \quad P(E_1 \cap E_2 \cap E_3^c | E_4) = \frac{1}{6}. \] Then, \( P(E_1 \cup E_2 \cup E_3 | E_4) \) is equal to

Let \( (X, Y) \) be a random vector with joint moment generating function \[ M(t_1, t_2) = \frac{1}{(1 - (t_1 + t_2))(1 - t_2)}, \quad -\infty<t_1, t_2<\min(1, 1 - t_2) \] Let \( Z = X + Y \). Then, \( \mathrm{Var}(Z) \) is equal to:

Let \( a_1 = 5 \) and define recursively \[ a_{n+1} = \frac{1}{3} \left(a_n\right)^{\frac{3}{4}}, \quad n \ge 1. \] Then, which of the following statements is TRUE?

Let \(X_1, X_2, ..., X_n\) be a random sample from an exponential distribution with probability density function \[ f(x; \theta) = \begin{cases} \theta e^{-\theta x}, & x>0,
0, & \text{otherwise,} \end{cases} \] where \(\theta \in (0, \infty)\) is unknown. Let \(\alpha \in (0,1)\) be fixed and let \(\beta\) be the power of the most powerful test of size \(\alpha\) for testing \(H_0: \theta = 1\) against \(H_1: \theta = 2\). Consider the critical region \[ R = \left\{ (x_1, x_2, ..., x_n) \in \mathbb{R}^n : \sum_{i=1}^n x_i>\frac{1}{2}\chi^2_{2n}(1-\alpha) \right\}, \] where for any \(\gamma \in (0,1)\), \(\chi^2_{2n}(\gamma)\) is a fixed point such that \( P(\chi^2_{2n}>\chi^2_{2n}(\gamma)) = \gamma. \) Then, the critical region \(R\) corresponds to the

Three coins have probabilities of head in a single toss as \(\frac{1}{4}\), \(\frac{1}{2}\), and \(\frac{3}{4}\) respectively. A player selects one coin at random and tosses it five times. The probability of obtaining two tails in five tosses is:

Let \( X_1, X_2, ..., X_n \) (\( n \ge 2 \)) be a random sample from \( U(\theta - 5, \theta + 5) \), where \( \theta \in (0, \infty) \) is unknown. Let \( T = \max(X_1, ..., X_n) \) and \( U = \min(X_1, ..., X_n) \). Then, which of the following statements is TRUE?

Let \( X \) be a continuous random variable having the moment generating function \[ M(t) = \frac{e^t - 1}{t}, \quad t \ne 0. \] Let \( \alpha = P(48X^2 - 40X + 3>0) \) and \( \beta = P((\ln X)^2 + 2\ln X - 3>0) \). Then, the value of \( \alpha - 2\ln \beta \) is equal to:

Let \( X \) and \( Y \) be random variables having chi-square distributions with 6 and 3 degrees of freedom respectively. Then, which of the following statements is TRUE?

Let \( X_1, X_2, ..., X_n \) (\( n \ge 3 \)) be a random sample from \( \text{Poisson}(\theta) \), where \( \theta>0 \) is unknown, and let \( T = \sum_{i=1}^{n} X_i \). Then, the uniformly minimum variance unbiased estimator (UMVUE) of \( e^{-2\theta}\theta^3 \) is:

If the series \(\sum_{n=1}^{\infty} a_n\) converges absolutely, then which of the following series diverges?

There are three urns labeled 1, 2, 3.
Urn 1: 2 white, 2 black; Urn 2: 1 white, 3 black; Urn 3: 3 white, 1 black.
Two coins are tossed independently, each with \(P(\text{head}) = 0.2\).
Urn 1 is selected if 2 heads occur, Urn 3 if 2 tails occur, otherwise Urn 2 is selected. A ball is drawn at random from the chosen urn. Find \[ P(\text{Urn 1 is selected} \mid \text{Ball drawn is white}) \]

Let \( \{X_n\}_{n \ge 1} \) be i.i.d. random variables distributed as \( N(0,1) \). Then find \[ \lim_{n \to \infty} P\left( \frac{\sum_{i=1}^{n} X_i^2 - 3n}{\sqrt{32n}} \le \sqrt{6} \right) \] is equal to:

Consider independent Bernoulli trials with success probability \( p = \frac{1}{3} \). The probability that three successes occur before four failures is:

Let \(X\) be a random variable with \[ f(x) = \frac{1}{2} e^{-|x|}, \quad -\infty<x<\infty \] Then, which of the following statements is FALSE?

Let \(X\) and \(Y\) be independent \(N(0,1)\) random variables and \(Z = \left|\frac{X}{Y}\right|\). Then, which of the following expectations is finite?

Let \(X\) be a \(U(0,1)\) random variable and \(Y = X^2\). If \(\rho\) is the correlation coefficient between \(X\) and \(Y\), then \(48\rho^2\) is equal to:

The value of the limit \[ \lim_{x \to 0} \frac{e^{-3x} - e^{x} + 4x}{5(1 - \cos x)} \] is equal to:

The value of the limit \[ \lim_{n \to \infty} \sum_{k=0}^{n} \binom{2n}{k} \frac{1}{4^n} \] is equal to: