List of top Questions asked in GATE ST- 2022

A random sample \( X_1, X_2, \dots, X_6 \) is taken from a Bernoulli distribution with the parameter \( \theta \). The null hypothesis \( H_0: \theta = \frac{1}{2} \) is to be tested against the alternative hypothesis \( H_1: \theta>\frac{1}{2} \), based on the statistic \( Y = \sum_{i=1}^{6} X_i \). If the value of \( Y \) corresponding to the observed sample values is 4, then the p-value of the test is ________?

Let \( X_1, X_2, \dots, X_n \) be a random sample from a distribution with cumulative distribution function \( F(x) \). Let the empirical distribution function of the sample be \( F_n(x) \). The classical Kolmogorov-Smirnov goodness of fit test statistic is given by \[ T_n = \sqrt{n} D_n = \sqrt{n} \sup_{-\infty<x<\infty} | F_n(x) - F(x) |. \] Consider the following statements: \begin{enumerate} \item The distribution of \( T_n \) is the same for all continuous underlying distribution functions \( F(x) \). \item \( D_n \) converges to 0 almost surely, as \( n \to \infty \). \end{enumerate} Which one of the following statements is/are true?

Let \( X_1, X_2, \dots, X_{20} \) be a random sample of size 20 from \( N_6(\mu, \Sigma) \), with det(\(\Sigma\)) \(\neq 0\), and suppose both \(\mu\) and \(\Sigma\) are unknown. Let \[ \bar{X} = \frac{1}{20} \sum_{i=1}^{20} X_i \quad \text{and} \quad S = \frac{1}{19} \sum_{i=1}^{20} (X_i - \bar{X})(X_i - \bar{X})^T. \] Consider the following two statements: \begin{enumerate} \item The distribution of \( 19S \) is \( W_6(19, \Sigma) \) (Wishart distribution of order 6 with 19 degrees of freedom). \item The distribution of \( (X_3 - \mu)^T S^{-1} (X_3 - \mu) \) is \( \chi^2_6 \) (Chi-square distribution with 6 degrees of freedom). \end{enumerate} Then which of the above statements is/are true?

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a function defined by \[ f(x, y) = \begin{cases} \frac{x^2 y}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \] Find the value of \( \frac{\partial f}{\partial x} \) at \( (0, 0) \).

Let \(M\) be a 2 \(\times\) 2 real matrix such that \((I + M)^{-1} = I - \alpha M\), where \(\alpha\) is a non-zero real number and \(I\) is the 2 \(\times\) 2 identity matrix. If the trace of the matrix \(M\) is 3, then the value of \(\alpha\) is

Let \(\{X(t)\}_{t\geq 0}\) be a linear pure death process with death rate \(\mu_i = 5i\), \(i = 0, 1, \dots, N\), \(N \geq 1\). Suppose that \(p_i(t) = P(X(t) = i)\). Then the system of forward Kolmogorov’s equations is

Let \( S^2 \) be the variance of a random sample of size \( n>1 \) from a normal population with an unknown mean \( \mu \) and an unknown finite variance \( \sigma^2>0 \). Consider the following statements:
(I) \( S^2 \) is an unbiased estimator of \( \sigma^2 \), and \( S \) is an unbiased estimator of \( \sigma \).
(II) \( \frac{n-1}{n} S^2 \) is a maximum likelihood estimator of \( \sigma^2 \), and \( \sqrt{\frac{n-1}{n}} S \) is a maximum likelihood estimator of \( \sigma \).
Which of the above statements is/are true?

Consider the following inequalities.
\text{(i) } 2x - 1 \(>\) 7
\text{(ii) } 2x - 9 \(<\)1
Which one of the following expressions below satisfies the above two inequalities?

In a class of five students P, Q, R, S and T, only one student is known to have copied in the exam. The disciplinary committee has investigated the situation and recorded the statements from the students as given below. Statement of P: R has copied in the exam.
Statement of Q: S has copied in the exam.
Statement of R: P did not copy in the exam.
Statement of S: Only one of us is telling the truth.
Statement of T: R is telling the truth.
The investigating team had authentic information that S never lies. Based on the information given above, the person who has copied in the exam is:

An art gallery engages a security guard to ensure that the items displayed are protected. The diagram below represents the plan of the gallery where the boundary walls are opaque. The location the security guard posted is identified such that all the inner space (shaded region in the plan) of the gallery is within the line of sight of the security guard.
If the security guard does not move around the posted location and has a 360° view, which one of the following correctly represents the set of ALL possible locations among the locations P, Q, R and S, where the security guard can be posted to watch over the entire inner space of the gallery?

Four points \( P(0, 1), Q(0, -3), R(-2, -1), \) and \( S(2, -1) \) represent the vertices of a quadrilateral. What is the area enclosed by the quadrilateral?

A sum of money is to be distributed among P, Q, R, and S in the proportion 5 : 2 : 4 : 3, respectively.
If R gets ₹1000 more than S, what is the share of Q (in ₹)?

A trapezium has vertices marked as P, Q, R, and S (in that order anticlockwise). The side PQ is parallel to side SR. Further, it is given that, PQ = 11 cm, QR = 4 cm, RS = 6 cm, and SP = 3 cm. What is the shortest distance between PQ and SR (in cm)?

The figure shows a grid formed by a collection of unit squares. The unshaded unit square in the grid represents a hole. What is the maximum number of squares without a "hole in the interior" that can be formed within the 4 $\times$ 4 grid using the unit squares as building blocks?

Mr. X speaks __________ Japanese __________ Chinese.