List of top Questions asked in GATE ST

Let \( X \) be a random variable having the probability density function

\[ f(x) = \begin{cases} \frac{3}{13} (1 - x)(9 - x), & 0<x<1, \\ 0, & \text{otherwise}. \end{cases} \]

Then

\[ \frac{4}{3} E[(X^2 - 15X + 27)] \]

equals _________ (round off to 2 decimal places).
Let \( \{X_n\}_{n \geq 0} \) be a time-homogeneous discrete time Markov chain with state space \( \{0, 1\} \) and transition probability matrix

\[ P = \begin{bmatrix} 0.25 & 0.75 \\ 0.75 & 0.25 \end{bmatrix}. \]

If \( P(X_0 = 0) = P(X_0 = 1) = 0.5 \), then

\[ \sum_{k=1}^{100} E[(X_{2k})^2] \]

equals _________ (round off to 2 decimal places).
If the marginal probability density function of the kth order statistic of a random sample of size 8 from a uniform distribution on [0, 2] is

\[ f(x) = \begin{cases} \frac{7}{32} x^6 (2 - x), & 0<x<2 \\ 0, & \text{otherwise} \end{cases} \]

then \( k \) equals _________ (round off to 2 decimal places).
Let \( X \) be a random variable having distribution function

\[ F(x) = \begin{cases} 0, & x<1 \\ a, & 1 \leq x<2 \\ \frac{c}{2}, & 2 \leq x<3 \\ 1, & x \geq 3 \end{cases} \]

where \( a \) and \( c \) are appropriate constants. Let \( A_n = \left[ 1 + \frac{1}{n}, 3 - \frac{1}{n} \right] \), \( n \geq 1 \), and \( A = \bigcup_{i=1}^{\infty} A_i \). If \( P(X \leq 1) = \frac{1}{2} \) and \( E(X) = \frac{5}{3} \), then \( P(X \in A) \) equals _________ (round off to 2 decimal places).
Let \( A \) be the 2 × 2 real matrix having eigenvalues 1 and -1, with corresponding eigenvectors \( \left[ \begin{array}{c} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{array} \right] \) and \( \left[ \begin{array}{c} \frac{-1}{2} \\ \frac{\sqrt{3}}{2} \end{array} \right] \), respectively. If \( A^{2021} = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \), then \( a + b + c + d \) equals _________ (round off to 2 decimal places).

Let 

and \( I_3 \) be the 3 × 3 identity matrix. Then the nullity of \( 5A(I_3 + A + A^2) \) equals _________

Let the joint distribution of \( (X,Y) \) be bivariate normal with mean vector 

and variance-covariance matrix 

, where \( -1<\rho<1 \). Let \( \Phi_\rho(0,0) = P(X \leq 0, Y \leq 0) \). Then the Kendall’s \( \tau \) coefficient between \( X \) and \( Y \) equals

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 

where \( \theta \in (0, \infty) \). Then the method of moments estimator of \( \theta \) equals

Let \( (X, Y) \) have a bivariate normal distribution with the joint probability density function \[ f_{X,Y}(x,y) = \frac{1}{\pi} e^{\left( \frac{3}{2} xy - \frac{25}{32} x^2 - 2 y^2 \right)} \] Then \[ E(XY) = _________ \text{ (round off to 2 decimal places).} \]
Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables each having uniform distribution on [0, 3]. Let \( Y \) be a random variable, independent of \( \{X_n\}_{n \geq 1} \), having probability mass function \[ P(Y = k) = \frac{1}{(e - 1) k!}, \quad k = 1, 2, \dots \] Then \[ P(\max(X_1, X_2, \dots, X_Y) \leq 1) = _________ \text{ (round off to 2 decimal places).} \]
Consider an amusement park where visitors are arriving according to a Poisson process with rate 1. Upon arrival, a visitor spends a random amount of time in the park and then departs. The time spent by the visitors are independent of one another, as well as of the arrival process, and have common probability density function \[ f(x) = \begin{cases} e^{-x}, & x>0,
0, & \text{otherwise}. \end{cases} \] If at a given time point, there are 10 visitors in the park and \( p \) is the probability that there will be exactly two more visitors before the next departure, then \[ p = _________ \text{ (round off to 2 decimal places).} \]
Let customers arrive at a departmental store according to a Poisson process with rate 10. Further, suppose that each arriving customer is either a male or a female with probability \( \frac{1}{2} \) each, independent of all other arrivals. Let \( N(t) \) denote the total number of customers who have arrived by time \( t \). Then which one of the following statements is NOT true?
Let \( Y \) follow \( N_8(0, I_8) \) distribution, where \( I_8 \) is the \( 8 \times 8 \) identity matrix. Let \( Y^T \Sigma_1 Y \) and \( Y^T \Sigma_2 Y \) be independent and follow central chi-square distributions with 3 and 4 degrees of freedom, respectively, where \( \Sigma_1 \) and \( \Sigma_2 \) are \( 8 \times 8 \) matrices and \( Y^T \) denotes transpose of \( Y \). Then which of the following statements is/are true? \[ P: \Sigma_1 \, \text{and} \, \Sigma_2 \, \text{are idempotent.} \quad Q: \Sigma_1 \Sigma_2 = 0, \, \text{where} \, 0 \, \text{is the} \, 8 \times 8 \, \text{zero matrix.} \]
Let \( A \) be a \( 3 \times 3 \) real matrix such that \( I_3 + A \) is invertible and let \[ B = (I_3 + A)^{-1}(I_3 - A), \] where \( I_3 \) denotes the \( 3 \times 3 \) identity matrix. Then which one of the following statements is true?
Let \( X \) be a random variable having Poisson distribution such that \( E(X^2) = 110 \). Then which one of the following statements is NOT true?
Let \( X \) be a random variable having uniform distribution on \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \). Then which one of the following statements is NOT true?
Let \( \Omega = \{1, 2, 3, \dots \} \) represent the collection of all possible outcomes of a random experiment with probabilities \( P(\{n\}) = a_n \) for \( n \in \Omega \). Then which one of the following statements is NOT true?
Let \( X_1, X_2, X_3 \) be three uncorrelated random variables with common variance \( \sigma^2<\infty \). Let \( Y_1 = 2X_1 + X_2 + X_3 \), \( Y_2 = X_1 + 2X_2 + X_3 \), and \( Y_3 = X_1 + X_2 + 2X_3 \). Then which of the following statements is/are true?
Let \( \{X_n\}_{n \geq 0} \) be a time-homogeneous discrete time Markov chain with either finite or countable state space \( S \). Then which one of the following statements is true?
Let \( X_1, X_2, X_3 \) be a random sample from a bivariate normal distribution with unknown mean vector \( \mu \) and unknown variance-covariance matrix \( \Sigma \), which is a positive definite matrix. The p-value corresponding to the likelihood ratio test for testing \[ H_0: \mu = 0 \quad \text{against} \quad H_1: \mu \neq 0 \] based on the realization \[ \left\{ \left( 1, 2 \right), \left( 4, -2 \right), \left( -5, 0 \right) \right\} \] of the random sample equals _________ (round off to 2 decimal places).
Let \( M \) be the collection of all \( 3 \times 3 \) real symmetric positive definite matrices. Consider the set \[ S = \left\{ A \in M : A^{50} - \frac{1}{4} A^{48} = 0 \right\}, \] where \( 0 \) denotes the \( 3 \times 3 \) zero matrix. Then the number of elements in \( S \) equals
Let \( A \) and \( B \) be two events such that \( P(B) = \frac{3}{4} \) and \( P(A \cup B^C) = \frac{1}{2} \). If \( A \) and \( B \) are independent, then \( P(A) \) equals _________ (round off to 2 decimal places).
A fair die is rolled twice independently. Let \( X \) and \( Y \) denote the outcomes of the first and second roll, respectively. Then \[ E(X + Y \mid (X - Y)^2 = 1) \] The value of \( E(X + Y \mid (X - Y)^2 = 1) \) is _________ (round off to 2 decimal places).
For \( \alpha>0 \), let \[ \{ X^{(\alpha)}_n \}_{n \geq 1} \text{ be a sequence of independent random variables such that} \quad P(X^{(\alpha)}_n = 1) = \frac{1}{n^{2\alpha}} = 1 - P(X^{(\alpha)}_n = 0). \] Let \( S = \{ \alpha>0 : X^{(\alpha)}_n \text{ converges to } 0 \text{ almost surely as } n \to \infty \}. \) \text{Then the infimum of \( S \) equals} _________ \text{ (round off to 2 decimal places).}
Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables each having uniform distribution on \( [0, 2] \). For \( n \geq 1 \), let \[ Z_n = - \log \left( \prod_{i=1}^{n} \left( 2 - X_i \right) \right)^{\frac{1}{n}}. \] Then, as \( n \to \infty \), the sequence \( \{Z_n\}_{n \geq 1} \) converges almost surely to _________ (round off to 2 decimal places).