List of top Statistics Questions

Let \( A \) be a \( 2 \times 2 \) real matrix such that \( AB = BA \) for all \( 2 \times 2 \) real matrices \( B \). If the trace of \( A \) equals 5, then the determinant of \( A \) (rounded off to two decimal places) equals ................

Let \( f: \mathbb{R}^2 ⇒ \mathbb{R} \) be defined by \( f(x, y) = xy \). Then the maximum value (rounded off to two decimal places) of \( f \) on the ellipse \( x^2 + 2y^2 = 1 \) equals ................

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables, each having mean 4 and variance 9. If \( Y_n = \frac{1}{n} \sum_{i=1}^{n} X_i \) for \( n \geq 1 \), then \[ \lim_{n \to \infty} E \left[ \left( \frac{Y_n - 4}{\sqrt{n}} \right)^2 \right] \text{ (in integer) equals:} \]

Let \( X_1, X_2, ...., X_n \) be a random sample of size \( n \) from a population having uniform distribution over the interval \( \left( \frac{1}{3}, \theta \right) \), where \( \theta>\frac{1}{3} \) is an unknown parameter. If \( Y = \max \{ X_1, X_2, ...., X_n \} \), then which one of the following statements is true?

Suppose that \( X_1, X_2, ...., X_n, Y_1, Y_2, ...., Y_n \) are independent and identically distributed random vectors each having \( N_p(\mu, \Sigma) \) distribution, where \( \Sigma \) is non-singular, \( p>1 \) and \( n>1 \). If \( \overline{X} = \frac{1}{n} \sum_{i=1}^n X_i \) and \( \overline{Y} = \frac{1}{n} \sum_{i=1}^n Y_i \), then which one of the following statements is true?

Suppose that $x$ is an observed sample of size 1 from a population with probability density function $f(.)$. Based on $x$, consider testing \[ H_0: f(y) = \frac{1}{\sqrt{2\pi}} e^{-y^2/2}, y \in \mathbb{R} \text{against} H_1: f(y) = \frac{1}{2} e^{-|y|}, y \in \mathbb{R}. \] Then which one of the following statements is true?

Let \( X \) be a random variable having Poisson distribution with mean \( \lambda>0 \). Then \( E \left( \frac{1}{X+1} \mid X>0 \right) \) equals:

Let \( \{X_n\}_{n \geq 1} \) and \( \{Y_n\}_{n \geq 1} \) be two sequences of random variables and \( X \) and \( Y \) be two random variables, all of them defined on the same probability space. Which one of the following statements is true?

Let \( A \) be a \( 3 \times 3 \) real matrix having eigenvalues \( 1, 0, \) and \( -1 \). If \( B = A^2 + 2A + I_3 \), where \( I_3 \) is the \( 3 \times 3 \) identity matrix, then which one of the following statements is true?

Let 𝑋1, 𝑋2 be a random sample from a distribution having a probability density function
\(f(x) =\begin{cases}   \frac{1}{θ}e^{\frac{y}{θ}}   & \quad \text{if }x >0,\\  0, & \quad Otherwise \end{cases}\)\(𝜃\)
where 𝜃∈(0, ∞) is an unknown parameter. For testing the null hypothesis 𝐻₀ : 𝜃=1 against 𝐻1∶ 𝜃≠1, consider a test that rejects 𝐻₀ for small observed values of the statistic \(𝑊 = \frac{𝑋_1+𝑋_2}{ 2}\) . If the observed values of 𝑋₁ and 𝑋₂ are 0.25 and 0.75, respectively, then the 𝑝-value equals___(round off to two decimal places)

Suppose that a fair coin is tossed repeatedly and independently. Let X denote the number of tosses required to obtain for the first time a tail that is immediately preceded by a head. Then E(X) and P(X > 4), respectively, are

Let X be a random variable having the probability density function
\(f(x)=\begin{cases} ax^2+b, & 0 \le x \le 3\\ 0, & \text{otherwise,} \end{cases}\)
where a and b are real constants, and \(P(X \ge 2)=\frac{2}{3}.\)
Then E(X) equals __________ (round off to 2 decimal places)

Suppose that four persons enter a lift on the ground floor of a building. There are seven floors above the ground floor and each person independently chooses her exit floor as one of these seven floors. If each of them chooses the topmost floor with probability \(\frac{1}{3}\) and each of the remaining floors with an equal probability, then the probability that no two of them exit at the same floor equals

\(\lim\limits_{n \rightarrow \infin}\frac{6}{n+2}\left\{(2+\frac{1}{n})^2+(2+\frac{2}{n})^2+...+(2+\frac{n-1}{2})^2\right\}\) equals

Let (X, Y) be a random vector having bivariate normal distribution with parameters E(X) = 0, Var(X) = 1, E(Y) = −1, Var(Y) = 4 and ρ(X, Y) = \(-\frac{1}{2}\) , where ρ(X, Y) denotes the correlation coefficient between X and Y. Then P(X + Y > 1 | 2X − Y = 1) equals

Let X be a random variable having binomial distribution with parameters n(> 1) and p(0 < p < 1). Then \(E(\frac{1}{1+X})\) equals

Let X be a random variable with the moment generating function
\(M(t)=\frac{1}{(1-4t)^5},t \lt\frac{1}{4}.\)
Then the lower bounds for P(X < 40), using Chebyshev’s inequality and Markov’s inequality, respectively, are

Let X₁ ,X₂ , … , X₁₆ be a random sample from a N(4μ, 1) distribution and Y₁ ,Y₂ , … , Y₈ be a random sample from a N(μ, 1) distribution, where μ ∈ \(\R\) is unknown. Assume that the two random samples are independent. If you are looking for a confidence interval for μ based on the statistic \(8\overline{X} + \overline{Y}\), where \(\overline{X}=\frac{1}{16}\sum^{16}_{i=1}X_i\) and \(\overline{Y}=\frac{1}{8}\sum^8_{i=1}Y_i\), then which one of the following statements is true ?

Let X₁,X₂ and X₃ be three independent and identically distributed random variables having U(0, 1) distribution. Then \(E[(\frac{\ln X_1}{\ln X_1 X_2 X_3})^2]\) equals

Let (X, Y) be a discrete random vector. Then which of the following statements is/are true ?

Let (X, Y) be a random vector having the joint moment generating function
\(M(t_1,t_2)=(\frac{1}{2}e^{-t_1}+\frac{1}{2}e^{t_1})^2(\frac{1}{2}+\frac{1}{2}et_2)^2,\ \ (t_1,t_2)\in \R^2.\)
Then P( |X + Y| = 2) equals __________ (round off to 2 decimal places)

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables having U(0, 1) distribution. Let Y_n = n min{X₁, X₂ , … , X_n}, n ≥ 1. If Y_n converges to Y in distribution, then the median of Y equals __________ (round off to 2 decimal places)

Let Ω = {1, 2, 3, … } be the sample space of a random experiment and suppose that all subsets of Ω are events. Further, let P be a probability function such that P({i}) > 0 for all i ∈ Ω. Then which of the following statements is/are true ?

Let f : \(\R→\R\) be the function defined by
\(f(x)=\begin{cases} \lim\limits_{h \rightarrow0}\frac{(x+h)\sin(\frac{1}{x}+h)-x\sin\frac{1}{x}}{h} & x \ne 0\\ 0, & x=0\end{cases}\)
Then which one of the following statements is NOT true ?

For n = 1, 2, 3, …, let the joint moment generating function of (X, Y_n) be
\(M_{X,Y_n}(t_1,t_2)=e^{\frac{t^2_1}{2}(1-2t_2)^{-\frac{n}{2}}}, t_1 \in \R,t_2 \lt \frac{1}{2}.\)
If \(T_n=\frac{\sqrt{n}X}{\sqrt{Y_n}},n \ge1,\) then which one of the following statements is true ?