List of top Questions asked in GATE ST- 2023

Two defective bulbs are present in a set of five bulbs. To remove the two defective bulbs, the bulbs are chosen randomly one by one and tested. If \( X \) denotes the minimum number of bulbs that must be tested to find out the two defective bulbs, then \( P(X = 3) \) (rounded off to two decimal places) 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?

The area of the region bounded by the parabola \( x = -y^2 \) and the line \( y = x + 2 \) equals

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?

Based only on the truth of the statement ‘Some humans are intelligent’, which one of the following options can be logically inferred with certainty?

The James Webb telescope, recently launched in space, is giving humankind unprecedented access to the depths of time by imaging very old stars formed almost 13 billion years ago. Astrophysicists and cosmologists believe that this odyssey in space may even shed light on the existence of dark matter. Dark matter is supposed to interact only via the gravitational interaction and not through the electromagnetic-, the weak- or the strong-interaction. This may justify the epithet “dark” in dark matter. Based on the above paragraph, which one of the following statements is FALSE?

Let \(a = 30!\), \(b = 50!\), and \(c = 100!\). Consider the following numbers: \(\log_a c\), \(\log_c a\), \(\log_b a\), \(\log_a b\)
Which one of the following inequalities is CORRECT?

“I have not yet decided what I will do this evening; I ................. visit a friend.”

Eject : Insert :: Advance : ............... (By word meaning)

A duck named Donald Duck says “All ducks always lie.” Based only on the information above, which one of the following statements can be logically inferred with certainty?