List of top Questions asked in GATE ST

Evaluate the limit: \[ \lim_{n \to \infty} \frac{(2^n + n 2^n \sin^2 \frac{n}{2})}{(2n - n \cos \frac{1}{n})} \] The value of the limit is _________ (round off to 2 decimal places).

Let \[ I = 4 \int_0^{\frac{1}{\sqrt{2}}} \int_0^x \frac{1}{\sqrt{x^2 + y^2}} \, dy \, dx \] Then the value of \( e^{l+\pi} \) is _________ (round off to 2 decimal places).

Let \( W(t) \) be a standard Brownian motion. Then the variance of \( W(1)W(2) \) equals

Let \( \{x_1, x_2, \dots, x_n\} \) be a realization of a random sample of size \( n (\geq 2) \) from a \( N(\mu, \sigma^2) \) distribution, where \( -\infty<\mu<\infty \) and \( \sigma>0 \). Which of the following statements is/are true?

Let \( X_1, X_2, \dots, X_n \) be a random sample of size \( n (\geq 2) \) from a \( N(0, \sigma^2) \) distribution. For a given \( \sigma>0 \), let \( f_\sigma \) denote the joint probability density function of \( (X_1, X_2, \dots, X_n) \) and \( S = \{ f_\sigma : \sigma>0 \} \). Let \( T_1 = \sum_{i=1}^{n} X_i^2 \) and \( T_2 = \left( \frac{1}{n} \sum_{i=1}^n X_i \right)^2 \). For any positive integer \( v \) and any \( \alpha \in (0, 1) \), let \( \chi^2_{v, \alpha} \) denote the \( (1 - \alpha) \)-th quantile of the central chi-square distribution with \( v \) degrees of freedom. Consider testing \( H_0: \sigma = 1 \) against \( H_1: \sigma>1 \) at level \( \alpha \). Then which one of the following statements is true?

Let \( X \) and \( Y \) be two random variables such that \( p_{11} + p_{10} + p_{01} + p_{00} = 1 \), where \( p_{ij} = P(X = i, Y = j) \), \( i, j = 0, 1 \). Suppose that a realization of a random sample of size 60 from the joint distribution of \( (X,Y) \) gives \( n_{11} = 10 \), \( n_{10} = 20 \), \( n_{01} = 20 \), \( n_{00} = 10 \), where \( n_{ij} \) denotes the frequency of \( (i,j) \) for \( i,j = 0,1 \). If the chi-square test of independence is used to test \[ H_0: p_{ij} = p_i p_j \text{ for } i,j = 0,1 \quad \text{against} \quad H_1: p_{ij} \neq p_i p_j \text{ for at least one pair } (i,j), \] where \( p_i = p_{i0} + p_{i1} \) and \( p_j = p_{0j} + p_{1j} \), then which one of the following statements is true?

Let \( X \) be a non-constant positive random variable such that \( E(X) = 9 \). Then which one of the following statements is true?