List of top Statistics Questions

Let $X_1, X_2, \dots, X_{10}$ be a random sample from $N(1,2)$ distribution. If $\bar{X} = \dfrac{1}{10} \sum_{i=1}^{10} X_i$ and $S^2 = \dfrac{1}{9} \sum_{i=1}^{10} (X_i - \bar{X})^2$, then $\text{Var}(S^2)$ equals

Consider a sequence of independent Bernoulli trials with probability of success in each trial being $\dfrac{1}{5}$. Then which of the following statements is/are TRUE?

Let $X_1, X_2, \ldots, X_n$ be a random sample from a distribution with probability density function \[ f_{\theta}(x) = \begin{cases} \theta (1 - x)^{\theta - 1}, & 0 < x < 1, \\ 0, & \text{otherwise}, \end{cases} \theta > 0. \] To test $H_0: \theta = 1$ against $H_1: \theta > 1$, the uniformly most powerful (UMP) test of size $\alpha$ would reject $H_0$ if

Let $X_1, X_2, \ldots, X_n$ be a random sample from $U(\theta - 0.5, \theta + 0.5)$ distribution, where $\theta \in \mathbb{R}$. If $X_{(1)} = \min(X_1, X_2, \ldots, X_n)$ and $X_{(n)} = \max(X_1, X_2, \ldots, X_n)$, then which one of the following estimators is NOT a maximum likelihood estimator (MLE) of $\theta$?

Consider a sequence of independent Bernoulli trials with probability of success in each trial being $\dfrac{1}{3}$. Let $X$ denote the number of trials required to get the second success. Then $P(X \ge 5)$ equals

Let $X_1, X_2, \dots, X_n$ be i.i.d. random variables having $N(\mu, \sigma^2)$ distribution, where $\mu \in \mathbb{R}$ and $\sigma > 0$. Define \[ W = \frac{1}{2n^2} \sum_{i=1}^{n} \sum_{j=1}^{n} (X_i - X_j)^2. \] Then $W$, as an estimator of $\sigma^2$, is

Let $\{X_n\}_{n \ge 1}$ be a sequence of i.i.d. random variables such that $E(X_i) = 1$ and $\text{Var}(X_i) = 1$. Then the approximate distribution of $\dfrac{1}{\sqrt{n}} \sum_{i=1}^n (X_{2i} - X_{2i-1})$, for large $n$, is

Let $X$ be a random variable having $U(0,10)$ distribution and $Y = X - [X]$, where $[X]$ denotes the greatest integer less than or equal to $X$. Then $P(Y > 0.25)$ equals ..............

Let $Z_1$ and $Z_2$ be i.i.d. $N(0,1)$ random variables. If $Y = Z_1^2 + Z_2^2$, then $P(Y > 4)$ equals

A packet contains 10 distinguishable firecrackers out of which 4 are defective. If three firecrackers are drawn at random (without replacement) from the packet, then the probability that all three firecrackers are defective equals

The mean and the standard deviation of weights of ponies in a large animal shelter are 20 kg and 3 kg, respectively. A pony is selected at random. Using Chebyshev's inequality, find the lower bound of the probability that its weight lies between 14 kg and 26 kg.

Let $E$ and $F$ be two events. Then which one of the following statements is NOT always TRUE?

Let $X$ be a random variable having Poisson(2) distribution. Then $E\left(\dfrac{1}{1+X}\right)$ equals

Let the Taylor polynomial of degree 20 for $ \frac{1}{(1-x)^3} $ at $ x = 0 $ be

\[ \sum_{n=0}^{20} a_n x^n \]

Then $ a_{15} $ equals

If

\[ P = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 0 & 1 \\ 0 & 0 & -1 \end{bmatrix} \quad \text{and} \quad 6P^{-1} = aI_3 + bP - P^2, \quad \text{then the ordered pair} \quad (a,b) \quad \text{is} \]

Let $ X_1, X_2, \dots, X_n $ be a random sample from a continuous distribution with the probability density function $ f(x) $. To test the hypothesis $ H_0: f(x) = e^{-x^2} $ against $ H_1: f(x) = e^{-2|x|} $, the rejection region of the most powerful size $ \alpha $ test is of the form, for some $ c>0 $,

Let $ P $ be a $ 2 \times 2 $ real matrix such that every non-zero vector in $ \mathbb{R}^2 $ is an eigenvector of $ P $. Suppose that $ \lambda_1 $ and $ \lambda_2 $ denote the eigenvalues of $ P $ and $ P \left[ \begin{array}{c} \sqrt{2} \\ \sqrt{3} \end{array} \right] = \left[ \begin{array}{c} 2 \\ t \end{array} \right] $ for some $ t \in \mathbb{R} $. Which of the following statements is (are) TRUE?

Let $ X $ be a random variable with the cumulative distribution function

\[ F(x) = \begin{cases} 0, & x < 0 \\ 1 + x^2, & 0 \leq x < 1 \\ \frac{10}{3} + x^2, & 1 \leq x < 2 \\ 1, & x \geq 2 \end{cases} \]

Which of the following statements is (are) TRUE?

Let $ X $ and $ Y $ be i.i.d. $ \text{Exp}(\lambda) $ random variables. If $ Z = \max\{X - Y, 0\} $, then which of the following statements is (are) TRUE?

Let the discrete random variables $ X $ and $ Y $ have the joint probability mass function

Which of the following statements is (are) TRUE?

Let $ X_1, X_2, \dots $ be a sequence of i.i.d. continuous random variables with the probability density function

If $ S_n = X_1 + X_2 + \dots + X_n $ and $ \bar{X}_n = \frac{S_n}{n} $, then the distributions of which of the following sequences of random variables converge(s) to a normal distribution with mean 0 and a finite variance?

Let $ X_1, X_2, \dots, X_n $ be a random sample from a continuous distribution with the probability density function for $ \lambda>0 $

To test the hypothesis $ H_0: \lambda = \frac{1}{2} $ against $ H_1: \lambda = \frac{3}{4} $ at the level $ \alpha $ (with $ 0<\alpha<1 $), which of the following statements is (are) TRUE?

Let $ f: [0, 2] \to \mathbb{R} $ be such that $ |f(x) - f(y)| \leq |x - y|^{4/3} $ for all $ x, y \in [0, 2] $. If $ \int_0^2 f(x) \, dx = \frac{2}{3} $, then

If

\[ \begin{pmatrix} \frac{\sqrt{5}}{3} & -\frac{2}{3} & c \\ \frac{2}{3} & \frac{\sqrt{5}}{3} & d \\ a & b & 1 \end{pmatrix} \]

is a real orthogonal matrix, then $ a^2 + b^2 + c^2 + d^2 $ equals ...............

Let $ 0, 1, 0, 0, 1 $ be the observed values of a random sample of size five from a discrete distribution with the probability mass function $ P(X = 1) = 1 - P(X = 0) = 1 - e^{-\lambda} $, where $ \lambda>0 $. The method of moments estimate (round off to 2 decimal places) of $ \lambda $ equals .............