List of top Questions asked in GATE ST

Let \(X\) and \(Y\) be independent and identically distributed normal random variables. If
\[ P(X+2Y\le3)=P(2X-Y\ge4), \]
then \(E(X)\) is

Let \(X_1,X_2\) be a random sample from the following probability density function
\[ f_\alpha(x)= \begin{cases} \alpha x^{\alpha-1}e^{-x^\alpha} & \text{if } x>0 \\ 0 & \text{otherwise,} \end{cases} \]
where \(\alpha\in(0,\infty)\) is an unknown parameter. Let
\[ X_{(1)}=\min\{X_1,X_2\} \quad\text{and}\quad X_{(2)}=\max\{X_1,X_2\}. \]
Then which of the following statements is correct?

Consider the polynomial \[ p(t)=t(t-1)(t-2) \] and define \[ F(x)=\int_{1/2}^{x}\frac{1}{p(t)}\,dt,\qquad x\in\left[\frac{1}{2},1\right). \] Which of the following statements is correct?

Let \(f:(-1,1)\to\mathbb{R}\) be a differentiable function. Consider the following statements:
(I) Suppose \(f(0)\geq0\), and \(f'(x)>0\) whenever \(f(x)=0\), for any \(x\geq0\). Then \(f(x)>0\), for any \(x>0\).
(II) Suppose \(f(0)\leq0\), and \(f'(x)>0\) whenever \(f(x)=0\), for any \(x\leq0\). Then \(f(x)<0\), for any \(x<0\).
Which of the following statements is correct?

In the 2020 summer Olympics' Javelin throw finals, Neeraj Chopra exhibited a spectacular performance to win the gold medal. The silver medal was won by Jakub Vadlejch and the bronze medal was won by Vitezlav Vesely. There were six rounds of throws with each athlete having one throw per round. The best of all the throws of each athlete is considered for the medal. Following were the observations about the throws:

(i) The first and second rounds were dominated by Neeraj Chopra with a gold medal performance in his second throw, while the other two athletes did not have any medal winning throws in these rounds.

(ii) The throws in the last round by both Jakub Vadlejch and Vitezlav Vesely were fouls and were not considered for scoring.

(iii) After four rounds, Vitezlav Vesely was in the second position and could not improve upon his best throw in the succeeding rounds.

(iv) In the fourth round, the throw by Jakub Vadlejch was the best in that round.

In which round did Vitezlav Vesely have his best throw?

Consider the following two subspaces of \(\mathbb{R}^4\) \[ W_1=\{(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0\} \] \[ W_2=\{(x_1,x_2,x_3,x_4):x_1+2x_2+3x_3+4x_4=0\}. \] Which of the following statements is correct?

An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice in succession and the number on the top face is recorded each time. The probability that the number appearing in the second roll is an integer multiple of the number appearing in the first roll is ______.

'When the teacher is in the room, all students stand silently.'
If the above statement is true, which one of the following statements is not necessarily true?

In Panel I of the figure below, the front view and top view of a structure are shown. Which one of the 3D structures shown in Panel II possesses the views shown in Panel I?

For positive real numbers \(S\) and \(K\), the function \(H_K(S)\) is defined as:
\[ H_K(S) = \max(S-K, 0). \]
The max function is defined as:
\[ \max(a,b) = \begin{cases} a, & \text{when } a > b \\ b, & \text{when } a \leq b \end{cases} \]
The graph below shows the plot of a function \(N(S)\) versus \(S\).
\(N(S)\) can be expressed as ______.

Consider a knock-out women's badminton singles tournament where there are no ties. The loser in each game is eliminated from the tournament. Every player plays until she is defeated or remains the last undefeated player. The last undefeated player is declared the winner of the tournament. If there are 64 players in the beginning of the tournament, how many games should be played in total to declare the winner of the tournament?

A student needs to enroll for a minimum of 60 credits. A student cannot enroll for more than 70 credits. The credits are divided amongst project and three distinct sets of courses namely, core courses, specialization courses, and elective courses. It is compulsory for a student to enroll for exactly 15 credits of core courses and exactly 20 credits of project. In addition, a student has to enroll for a minimum of 10 credits of specialization courses. The maximum credits of elective courses that a student can enroll for is ______

Combinatorics deals with problems involving counting. For example, "How many distinct arrangements of N distinct objects in M spaces on a circle are possible?" is a typical problem in combinatorics. This kind of counting is sometimes used in the modeling of several physical phenomena. Often, in such models, the different combinatorial possibilities are assigned probability values. Assigning probabilities enables the computation of the average values of physical quantities.

Consider the following statements:

P: Combinatorics is always invoked in the modeling of physical phenomena.

Q: Modeling some physical phenomena involves assigning probabilities to combinatorial possibilities in order to compute average values of physical quantities.

Based on the passage above, what can be inferred about statements P and Q?

The figure shows two 4-tile patterns.

Either one or both of the patterns can be used any number of times and in any orientation to construct a new pattern. Which one of the options below cannot be constructed by using only these two 4-tile patterns, assuming there are no overlaps among them?

The antonym of the word protagonist is ________.

Let \[ I = \pi^2 \int_0^1 \int_0^1 y^2 \cos \pi(1 + xy) \, dx \, dy. \] The value of \( I \) is equal to _________ (answer in integer).

Let \( (X_1, X_2, X_3)^T \) have the following distribution \[ N_3 \left( \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & 0.4 & 0 \\ 0.4 & 1 & 0.6 \\ 0 & 0.6 & 1 \end{pmatrix} \right). \] Then the value of the partial correlation coefficient between \( X_1 \) and \( X_2 \) given \( X_3 \) is ___________ (rounded off to two decimal places).

Let \( X \sim \text{Bin}(3, \theta) \), where \( \theta \in (0,1) \) is an unknown parameter. For testing \[ H_0: \frac{1}{4} \leq \theta \leq \frac{3}{4} \quad \text{against} \quad H_1: \theta < \frac{1}{4} \quad \text{or} \quad \theta > \frac{3}{4}, \] consider the test \[ \phi(x) = \begin{cases} 1 & \text{if } x \in \{0, 3\}, \\ 0 & \text{if } x \in \{1, 2\}. \end{cases} \] The size of the test \( \phi \) is __________ (rounded off to two decimal places).

Let \( X_1, X_2 \) be a random sample from a population having probability density function \[ f_{\theta}(x) = \begin{cases} e^{(x-\theta)} & \text{if } -\infty < x \leq \theta, \\ 0 & \text{otherwise}, \end{cases} \] where \( \theta \in \mathbb{R} \) is an unknown parameter. Consider testing \( H_0: \theta \geq 0 \) against \( H_1: \theta < 0 \) at level \( \alpha = 0.09 \). Let \( \beta(\theta) \) denote the power function of a uniformly most powerful test. Then \( \beta(\log_e 0.36) \) equals ________ (rounded off to two decimal places).

Let \( X_1, X_2, \dots, X_7 \) be a random sample from a population having the probability density function \[ f(x) = \frac{1}{2} \lambda^3 x^2 e^{-\lambda x}, \quad x>0, \] where \( \lambda>0 \) is an unknown parameter. Let \( \hat{\lambda} \) be the maximum likelihood estimator of \( \lambda \), and \( E(\hat{\lambda} - \lambda) = \alpha \lambda \) be the corresponding bias, where \( \alpha \) is a real constant. Then the value of \( \frac{1}{\alpha} \) equals __________ (answer in integer).

Let \( \{ X_n \}_{n \geq 1} \) be a sequence of independent random variables with \[ \Pr(X_n = -\frac{1}{2^n}) = \Pr(X_n = \frac{1}{2^n}) = \frac{1}{2}, \quad \forall n \in \mathbb{N}. \] Suppose that \( \sum_{i=1}^{n} X_i \) converges to \( U \) as \( n \to \infty \). Then \( 6 \Pr(U \leq \frac{2}{3}) \) equals ___________ (answer in integer).

The moment generating functions of three independent random variables \( X, Y, Z \) { are respectively given as:} \[ M_X(t) = \frac{1}{9}(2 + e^t)^2, \quad t \in \mathbb{R}, \] \[ M_Y(t) = e^{e^t - 1}, \quad t \in \mathbb{R}, \] \[ M_Z(t) = e^{2(e^t - 1)}, \quad t \in \mathbb{R}. \] {Then } \( 10 \cdot \Pr(X>Y + Z) \) equals __________ (rounded off to two decimal places).

Let \[ P = \begin{pmatrix} 0 & 1 & 1 & 1 & 1 \\ -1 & 0 & 1 & 1 & 1 \\ -1 & -1 & 0 & 1 & 1 \\ -1 & -1 & -1 & 0 & 1 \\ -1 & -1 & -1 & -1 & 0 \end{pmatrix} \] If \( \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5 \) are eigenvalues of \( P \), then \( \prod_{i=1}^{5} \lambda_i = \) ________ (answer in integer).

Let \( \{ W(t) : t \geq 0 \} \) be a standard Brownian motion. Then \[ E\left( (W(2) + W(3))^2 \right) \] {equals __________ (answer in integer).}