List of top Standard Distributions Questions on Random Variables asked in GATE ST

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables each having probability density function

\[ f(x) = \begin{cases} e^{-x}, & x>0 \\ 0, & \text{otherwise} \end{cases} \]

Let \( X_{(n)} = \max\{ X_1, X_2, \dots, X_n \} \) for \( n \geq 1 \). If \( Z \) is the random variable to which

\[ \{ X_{(n)} - \log n \}_{n \geq 1} \]

converges in distribution, as \( n \to \infty \), then the median of \( Z \) equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Let \( \{X_n\}_{n \geq 1} \) be a sequence of independent and identically distributed random variables having common distribution function \( F(x) \). Let \( a<b \) be two real numbers such that \( F(x) = 0 \) for all \( x \leq a \), \( 0<F(x)<1 \) for all \( a<x<b \), and \( F(x) = 1 \) for all \( x \geq b \). Let \( S_n(x) \) be the empirical distribution function at \( x \) based on \( X_1, X_2, \dots, X_n \), \( n \geq 1 \). Then which one of the following statements is NOT true?

Let \( X_1, X_2, \dots, X_n \) be a random sample of size \( n (\geq 2) \) from a uniform distribution on \( [-\theta, \theta] \), where \( \theta \in (0, \infty) \). Let \( X_{(1)} = \min\{ X_1, X_2, \dots, X_n \} \) and \( X_{(n)} = \max\{ X_1, X_2, \dots, X_n \} \). Then which of the following statements is/are true?

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 \[ P(\max(X_1, X_2, \dots, X_Y) \leq 1) = _________ \text{ (round off to 2 decimal places).} \]

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 \( \{X_n\}_{n \geq 1} \) 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).

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?