List of top Questions asked in GATE ST

Consider the following sentences:
(i) I woke up from sleep.
(ii) I wok up from sleep.
(iii) I was woken up from sleep.
(iv) I was wokened up from sleep.
Which of the above sentences are grammatically CORRECT?
Nostalgia is to anticipation as _________ is to _________. Which one of the following options maintains a similar logical relation in the above sentence?

The least number of squares that must be added so that the line P-Q becomes the line of symmetry is 

p and q are positive integers and \[ \frac{p}{q} + \frac{q}{p} = 3, \] then, \[ \frac{p^2}{q^2} + \frac{q^2}{p^2} = \]

The current population of a city is 11,02,500. If it has been increasing at the rate of 5% per annum, what was its population 2 years ago? 
 

Consider the multiple linear regression model

\[ Y_i = \beta_0 + \beta_1 x_{1,i} + \beta_2 x_{2,i} + \dots + \beta_{22} x_{22,i} + \epsilon_i, \quad i = 1, 2, \dots, 123, \]

where, for \( j = 0, 1, 2, \dots, 22 \), \( \beta_j \)'s are unknown parameters and \( \epsilon_i \)'s are independent and identically distributed \( N(0, \sigma^2) \), \( \sigma>0 \), random variables. If the sum of squares due to regression is 338.92, the total sum of squares is 522.30 and \( R^2_{\text{adj}} \) denotes the value of adjusted \( R^2 \), then

\[ 100 R^2_{\text{adj}} = \underline{\hspace{2cm}} \]

(round off to 2 decimal places).
Let \( X_1, X_2, \dots, X_{10} \) be a random sample from a probability density function

\[ f_\theta(x) = f(x - \theta), \quad -\infty<x<\infty, \]

where \( -\infty<\theta<\infty \) and \( f(-x) = f(x) \) for \( -\infty<x<\infty \). For testing

\[ H_0: \theta = 1.2 \quad \text{against} \quad H_1: \theta \neq 1.2, \]

let \( T^+ \) denote the Wilcoxon Signed-rank test statistic. If \( \eta \) denotes the probability of the event \( \{T^+<50\} \) under \( H_0 \), then \( 32\eta \) equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Let \( X \) be a discrete random variable with probability mass function \( p \in \{p_0, p_1\} \), where 

To test \( H_0 : p = p_0 \) against \( H_1 : p = p_1 \), the power of the most powerful test of size 0.05, based on \( X \), equals _________ 

Let a random sample of size 100 from a normal population with unknown mean \( \mu \) and variance 9 give the sample mean 5.608. Let \( \Phi(\cdot) \) denote the distribution function of the standard normal random variable. If \( \Phi(1.96) = 0.975 \), \( \Phi(1.64) = 0.95 \), and the uniformly most powerful unbiased test based on sample mean is used to test \[ H_0: \mu = 5.02 \quad \text{against} \quad H_1: \mu \neq 5.02, \] then the p-value equals 

Let \( \{0.90, 0.50, 0.01, 0.95\} \) be a realization of a random sample of size 4 from the probability density function

\[ f(x) = \begin{cases} \frac{\theta}{(1-\theta)} x^{(2\theta-1)/(1-\theta)}, & 0<x<1 \\ 0, & \text{otherwise} \end{cases} \]

where \( 0.5 \leq \theta<1 \). Then the maximum likelihood estimate of \( \theta \) based on the observed sample equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).
Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables each having probability density function

\[ f(x) = \begin{cases} e^{-x}, & x>0 \\ 0, & \text{otherwise} \end{cases} \]

Let \( X_{(n)} = \max\{ X_1, X_2, \dots, X_n \} \) for \( n \geq 1 \). If \( Z \) is the random variable to which

\[ \{ X_{(n)} - \log n \}_{n \geq 1} \]

converges in distribution, as \( n \to \infty \), then the median of \( Z \) equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).
Let \( X \) be a random variable having the moment generating function

\[ M(t) = \frac{e^t - 1}{t(1 - t)}, \quad t<1. \]

Then

\[ P(X>1) = \underline{\hspace{2cm}} \]

(round off to 2 decimal places).
Let

\[ A = \begin{bmatrix} a & u_1 & u_2 & u_3 \end{bmatrix}, \quad B = \begin{bmatrix} b & u_1 & u_2 & u_3 \end{bmatrix}, \quad C = \begin{bmatrix} u_2 & u_3 & u_1 & a + b \end{bmatrix}. \]

Let \( \det(A), \det(B), \det(C) \) denote the determinants of the matrices \( A \), \( B \), and \( C \), respectively. If

\[ \det(A) = 6, \quad \det(B) = 2, \quad \text{then } \det(A + B) - \det(C) = \underline{\hspace{2cm}}. \]

Let \( f: \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) be defined by \[ f(x, y) = 8x^2 - 2y, \text{ where } \mathbb{R} \text{ denotes the set of all real numbers.} \] If \( M \) and \( m \) denote the maximum and minimum values of \( f \), respectively, on the set \[ \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}, \] then M - m = _________ (round off to 2 decimal places

Let the joint distribution of random variables \( X_1, X_2, X_3 \) and \( X_4 \) be \( N_4(\mu, \Sigma) \), where

Then which one of the following statements is true?

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables having common distribution function \( F(x) \). Let \( a<b \) be two real numbers such that \( F(x) = 0 \) for all \( x \leq a \), \( 0<F(x)<1 \) for all \( a<x<b \), and \( F(x) = 1 \) for all \( x \geq b \). Let \( S_n(x) \) be the empirical distribution function at \( x \) based on \( X_1, X_2, \dots, X_n \), \( n \geq 1 \). Then which one of the following statements is NOT true?
Let \( X_1, X_2, \dots, X_n \) be a random sample of size \( n (\geq 2) \) from a uniform distribution on \( [-\theta, \theta] \), where \( \theta \in (0, \infty) \). Let \( X_{(1)} = \min\{ X_1, X_2, \dots, X_n \} \) and \( X_{(n)} = \max\{ X_1, X_2, \dots, X_n \} \). Then which of the following statements is/are true?
Let \( X_1, X_2, \dots, X_n \) be a random sample of size \( n \geq 2 \) from a distribution having the probability density function

\[ f(x; \theta) = \begin{cases} \frac{1}{\theta} e^{-\frac{x}{\theta}}, & x>0 \\ 0, & \text{otherwise} \end{cases} \]

where \( \theta \in (0, \infty) \). Let \( X_{(1)} = \min\{ X_1, X_2, \dots, X_n \} \) and \( T = \sum_{i=1}^{n} X_i \). Then \( E(X_{(1)} \mid T) \) equals

Let 

be the order statistics corresponding to a random sample of size 5 from a uniform distribution on \( [0, \theta] \), where \( \theta \in (0, \infty) \). Then which of the following statements is/are true?} 
P: \( 3X_{(2)} \) is an unbiased estimator of \( \theta \). 
Q: The variance of \( E[2X_{(3)} \mid X_{(5)}] \) is less than or equal to the variance of \( 2X_{(3)} \).

Let \( (X, Y) \) have the joint probability density function

\[ f_{X,Y}(x,y) = \begin{cases} \frac{4}{(x + y)^3}, & x>1, y>1 \\ 0, & \text{otherwise} \end{cases} \]

Then which one of the following statements is NOT true?
Let \( f: \mathbb{R} \to \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \) and \( f'(x) + 2f(x)>0 \) for all \( x \in \mathbb{R} \), where \( f' \) denotes the derivative of \( f \) and \( \mathbb{R} \) denotes the set of all real numbers. Then which one of the following statements is true?

Let \( Y_i = \alpha + \beta x_i + \epsilon_i \), where \( x_i \)'s are fixed covariates, \( \alpha \) and \( \beta \) are unknown parameters, and \( \epsilon_i \)'s are independent and identically distributed random variables with mean zero and finite variance. Let \( \hat{\alpha} \) and \( \hat{\beta} \) be the ordinary least squares estimators of \( \alpha \) and \( \beta \), respectively. Given the following observations: 

The value of \( \hat{\alpha} + \hat{\beta} \) equals _________ (round off to 2 decimal places).

Let \( f: [0, \infty) \to \mathbb{R} \) be a function, where \( \mathbb{R} \) denotes the set of all real numbers. Then which one of the following statements is true?
Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by

\[ f(x) = \begin{cases} x^3 \sin x, & \text{if } x = 0 \text{ or } x \text{ is irrational}, \\ \frac{1}{q^3}, & \text{if } x = \frac{p}{q},\; p \in \mathbb{Z} \setminus \{0\},\; q \in \mathbb{N},\; \text{and } \gcd(p,q) = 1, \end{cases} \]

where \( \mathbb{R} \) denotes the set of all real numbers, \( \mathbb{Z} \) denotes the set of all integers, \( \mathbb{N} \) denotes the set of all positive integers, and \( \gcd(p,q) \) denotes the greatest common divisor of \( p \) and \( q \). Then which one of the following statements is true?
Let \( (Y, X_1, X_2) \) be a random vector with mean vector

\[ \begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix} \]

and covariance matrix

\[ \begin{pmatrix} 10 & 0.5 & -0.5 \\ 0.5 & 7 & 1.5 \\ -0.5 & 1.5 & 2 \end{pmatrix} \]

Then the value of the multiple correlation coefficient between \( Y \) and its best linear predictor on \( X_1 \) and \( X_2 \) equals _________ (round off to 2 decimal places).