List of top Questions asked in GATE ST

Suppose \( X_1, X_2, \dots, X_n, \dots \) are independent exponential random variables with the mean \( \frac{1}{2} \). Let the notation \( i.o. \) denote "infinitely often." Then which of the following is/are true?

Let \( \{ X_n \}, n \geq 1 \), be a sequence of random variables with the probability mass functions \[ p_{X_n}(x) = \begin{cases} \frac{n}{n+1}, & x = 0 \\ \frac{1}{n+1}, & x = n \\ 0, & \text{elsewhere} \end{cases} \] Let \( X \) be a random variable with \( P(X = 0) = 1 \). Then which of the following statements is/are true?

Let \(M\) be any \(3\times3\) symmetric matrix with eigenvalues \(1,2,3\). Let \(N\) be any \(3\times3\) matrix with real eigenvalues such that \(MN+N^{T}M=3I\), where \(I\) is the \(3\times3\) identity matrix. Which of the following cannot be eigenvalue(s) of the matrix \(N\)?

Let \( M \) be a \( 3 \times 2 \) real matrix having a singular value decomposition as \( M = U S V^T \), where the matrix \( S = \begin{bmatrix} \sqrt{3} & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}^T \), \( U \) is a \( 3 \times 3 \) orthogonal matrix, and \( V \) is a \( 2 \times 2 \) orthogonal matrix. Then which of the following statements is/are true?

Consider the following infinite series: \[ S_1 := \sum_{n=0}^{\infty} (-1)^n \frac{n}{n^2 + 4} \quad \text{and} \quad S_2 := \sum_{n=0}^{\infty} (-1)^n \sqrt{n^2 + 1 - n}. \] Which of the above series is/are conditionally convergent?

Let \( (3, 6)^T, (4, 4)^T, (5, 7)^T \) and \( (4, 7)^T \) be four independent observations from a bivariate normal distribution with the mean vector \( \mu \) and the covariance matrix \( \Sigma \). Let \( \hat{\mu} \) and \( \hat{\Sigma} \) be the maximum likelihood estimates of \( \mu \) and \( \Sigma \), respectively, based on these observations. Then \( \hat{\Sigma} \hat{\mu} \) is equal to

Let \( X = \begin{pmatrix} X_1 \\ X_2 \\ X_3 \end{pmatrix} \) follow \( N_3(\mu, \Sigma) \) with \( \mu = \begin{pmatrix} 2 \\ -3 \\ 2 \end{pmatrix} \) and \( \Sigma = \begin{pmatrix} 4 & -1 & 1 \\ -1 & 2 & a \\ 1 & a & 2 \end{pmatrix} \), where \( a \in \mathbb{R} \). Suppose that the partial correlation coefficient between \( X_2 \) and \( X_3 \), keeping \( X_1 \) fixed, is \( \frac{5}{7} \). Then \( a \) is equal to

If the line \( y = \alpha x, \alpha \geq \sqrt{2} \), divides the area of the region \[ R := \{(x, y) \in \mathbb{R}^2 \mid 0 \leq x \leq \sqrt{y}, 0 \leq y \leq 2\} \] into two equal parts, then the value of \( \alpha \) is equal to

Let \( (X, Y, Z) \) be a random vector with the joint probability density function \[ f_{X,Y,Z}(x, y, z) = \begin{cases} \frac{1}{3}(2x + 3y + z), & 0<x<1, 0<y<1, 0<z<1, \\ 0, & \text{elsewhere}. \end{cases} \] Then which one of the following points is on the regression surface of \( X \) on \( (Y, Z) \)?

A random sample \( X \) of size one is taken from a distribution with the probability density function \[ f(x; \theta) = \begin{cases} \frac{2x}{\theta^2}, & 0<x<\theta, \\ 0, & \text{elsewhere.} \end{cases} \] If \( \frac{X}{\theta} \) is used as a pivot for obtaining the confidence interval for \( \theta \), then which one of the following is an 80% confidence interval (confidence limits rounded off to three decimal places) for \( \theta \) based on the observed sample value \( x = 10 \)?

Let \( X_1, X_2, \dots, X_7 \) be a random sample from a normal population with mean 0 and variance \( \theta>0 \). Let \[ K = \frac{X_1^2 + X_2^2 + \dots + X_7^2}{X_1^2 + X_2^2 + \dots + X_7^2}. \] Consider the following statements: \begin{enumerate} \item The statistics \( K \) and \( X_1^2 + X_2^2 + \dots + X_7^2 \) are independent. \item \( \frac{7K}{2} \) has an \( F \)-distribution with 2 and 7 degrees of freedom. \item \( E(K^2) = \frac{8}{63} \). \end{enumerate} Then which of the above statements is/are true?

Let \( X \) be a random variable with the probability density function \[ f(x) = \begin{cases} c(x - \lfloor x \rfloor), & 0<x<3, \\ 0, & \text{elsewhere}, \end{cases} \] where \( c \) is a constant and \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). If \[ A = \left[ \frac{1}{2}, 2 \right], \text{ then } P(X \in A) \, \text{(rounded off to two decimal places) is equal to} \, ________. \]

Let \( X \) and \( Y \) be two random variables such that the moment generating function of \( X \) is \( M(t) \) and the moment generating function of \( Y \) is \[ H(t) = \left( \frac{3}{4} e^{2t} + \frac{1}{4} \right) M(t), \] where \( t \in (-h, h), h>0 \). If the mean and the variance of \( X \) are \( \frac{1}{2} \) and \( \frac{1}{4} \), respectively, then the variance of \( Y \) (in integer) is equal to ________

Let \( X_i, i = 1, 2, \dots, n \), be i.i.d. random variables with the probability density function \[ f_X(x) = \begin{cases} \frac{1}{\sqrt{2 \Gamma\left( \frac{1}{6} \right)}} x^{-\frac{5}{6}} e^{-\frac{x}{8}}, & 0<x<\infty, \\ 0, & \text{elsewhere}, \end{cases} \] where \( \Gamma(\cdot) \) denotes the gamma function. Also, let \( \bar{X}_n = \frac{1}{n} (X_1 + X_2 + \cdots + X_n) \). If \[ \sqrt{n} \left( \bar{X}_n - \mathbb{E}[\bar{X}_n] \right) \xrightarrow{d} N(0, \sigma^2), \] then \( \sigma^2 \) (rounded off to two decimal places) is equal to ________

A random sample of size 4 is taken from the distribution with the probability density function \[ f(x; \theta) = \begin{cases} \frac{2(\theta - x)}{\theta^2}, & 0<x<\theta, \\ 0, & \text{elsewhere}. \end{cases} \] If the observed sample values are 6, 5, 3, 6, then the method of moments estimate (in integer) of the parameter \( \theta \), based on these observations, is ________

A company sometimes stops payments of quarterly dividends. If the company pays the quarterly dividend, the probability that the next one will be paid is 0.7. If the company stops the quarterly dividend, the probability that the next quarterly dividend will not be paid is 0.5. Then the probability (rounded off to three decimal places) that the company will not pay quarterly dividend in the long run is ________

Let \( X_1, X_2, \dots, X_8 \) be a random sample taken from a distribution with the probability density function \[ f_X(x) = \begin{cases} \frac{x}{8}, & 0<x<4, \\ 0, & \text{elsewhere}. \end{cases} \] Let \( F_8(x) \) be the empirical distribution function of the sample. If \( \alpha \) is the variance of \( F_8(2) \), then 128\( \alpha \) (in integer) is equal to ________

Let \( M \) be a 3 \(\times\) 3 real symmetric matrix with eigenvalues \(-1, 1, 2\) and the corresponding unit eigenvectors \( u, v, w \), respectively. Let \( x \) and \( y \) be two vectors in \( \mathbb{R}^3 \) such that \[ Mx = u + 2(v + w) \quad \text{and} \quad M^2 y = u - (v + 2w). \] Considering the usual inner product in \( \mathbb{R}^3 \), the value of \( |x + y|^2 \), where \( |x + y| \) is the length of the vector \( x + y \), is

Let \(\{a_n\}_{n=1}^\infty\) be a sequence of positive real numbers satisfying \[ \frac{8}{a_{n+1}} = \frac{7}{a_n} + \frac{a_n^2}{343}, \quad n \geq 1, \] with \(a_1 = 3\) and \(a_n<7\) for all \(n \geq 2\).
Consider the following statements: \begin{itemize} \item [(I)] \(\{a_n\}\) is monotonically increasing. \item [(II)] \(\{a_n\}\) converges to a value in the interval \([3, 7]\). \end{itemize} Then which of the above statements is/are true?

Let \( M \) be any square matrix of arbitrary order \( n \) such that \( M^2 = 0 \) and the nullity of \( M \) is 6. Then the maximum possible value of \( n \) (in integer) is ________

Consider the usual inner product in \( \mathbb{R}^4 \). Let \( u \in \mathbb{R}^4 \) be a unit vector orthogonal to the subspace \[ S = \{(x_1, x_2, x_3, x_4)^T \in \mathbb{R}^4 \mid x_1 + x_2 + x_3 + x_4 = 0 \}. \] If \( v = (1, -2, 1, 1)^T \), and the vectors \( u \) and \( v - \alpha u \) are orthogonal, then the value of \( \alpha^2 \) (rounded off to two decimal places) is equal to ________

Let \( \{ B(t) \}_{t \geq 0} \) be a standard Brownian motion and let \( \Phi(\cdot) \) be the cumulative distribution function of the standard normal distribution. If \[ P\left( \left( B(2) + 2B(3) \right)>1 \right) = 1 - \Phi\left( \frac{1}{\sqrt{\alpha}} \right), \, \alpha>0, \] then the value of \( \alpha \) (in integer) is equal to ________

Let \( X \) and \( Y \) be two independent exponential random variables with \( E(X^2) = \frac{1}{2} \) and \( E(Y^2) = \frac{2}{9} \). Then \( P(X<2Y) \) (rounded off to two decimal places) is equal to ________

Let \( X_i, i = 1, 2, \dots, n \), be i.i.d. random variables from a normal distribution with mean 1 and variance 4. Let \( S_n = X_1^2 + X_2^2 + \dots + X_n^2 \). If \( \text{Var}(S_n) \) denotes the variance of \( S_n \), then the value of \[ \lim_{n \to \infty} \left( \frac{\text{Var}(S_n)}{n} - \left( \frac{E(S_n)}{n} \right)^2 \right) \quad \text{(in integer) is equal to} \, ________. \]

At a telephone exchange, telephone calls arrive independently at an average rate of 1 call per minute, and the number of telephone calls follows a Poisson distribution. Five time intervals, each of duration 2 minutes, are chosen at random. Let \( p \) denote the probability that in each of the five time intervals at most 1 call arrives at the telephone exchange. Then \( e^{10} p \) (in integer) is equal to ________