List of top Standard Distributions Questions

Let \( X \) be a random variable with the probability mass function \( p_X(x) = \binom{x-1}{3} \left( \frac{3}{4} \right)^{x-1} \left( \frac{1}{4} \right) \), for \( x = 1, 2, 3, \dots \). Then the value of \[ \sum_{n=0}^{\infty} P(n<X \leq n + 3) \quad \text{(rounded off to two decimal places) is equal to} ________. \]

Let a random sample of size 100 from a normal population with unknown mean \( \mu \) and variance 9 give the sample mean 5.608. Let \( \Phi(\cdot) \) denote the distribution function of the standard normal random variable. If \( \Phi(1.96) = 0.975 \), \( \Phi(1.64) = 0.95 \), and the uniformly most powerful unbiased test based on sample mean is used to test \[ H_0: \mu = 5.02 \quad \text{against} \quad H_1: \mu \neq 5.02, \] then the p-value equals 

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 the joint distribution of random variables \( X_1, X_2, X_3 \) and \( X_4 \) be \( N_4(\mu, \Sigma) \), where

Then which one of the following statements is true?

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 \( X_1, X_2, \dots, X_n \) be a random sample of size \( n \geq 2 \) from a distribution having the probability density function

\[ f(x; \theta) = \begin{cases} \frac{1}{\theta} e^{-\frac{x}{\theta}}, & x>0 \\ 0, & \text{otherwise} \end{cases} \]

where \( \theta \in (0, \infty) \). Let \( X_{(1)} = \min\{ X_1, X_2, \dots, X_n \} \) and \( T = \sum_{i=1}^{n} X_i \). Then \( E(X_{(1)} \mid T) \) equals

Let 

be the order statistics corresponding to a random sample of size 5 from a uniform distribution on \( [0, \theta] \), where \( \theta \in (0, \infty) \). Then which of the following statements is/are true?} 
P: \( 3X_{(2)} \) is an unbiased estimator of \( \theta \). 
Q: The variance of \( E[2X_{(3)} \mid X_{(5)}] \) is less than or equal to the variance of \( 2X_{(3)} \).

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 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 \( \{X_n\}_{n \geq 1} \) 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 \( (X, Y) \) 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 \( Y \) 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 \( X \) 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 \( X \) be a random variable having Poisson distribution such that \( E(X^2) = 110 \). Then which one of the following statements is NOT true?
Let \( X_1, X_2, X_3 \) 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 \( \{X_n\}_{n \geq 1} \) 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 \( \{0, 2\} \) be a realization of a random sample of size 2 from a binomial distribution with parameters 2 and \( p \), where \( p \in (0, 1) \). To test \( H_0: p = \frac{1}{2} \), the observed value of the likelihood ratio test statistic equals _________ (round off to 2 decimal places).
Let \( X_1, X_2, X_3, Y_1, Y_2, Y_3, Y_4 \) be independent random vectors such that \( X_i \) follows \( N_4(0, \Sigma_1) \) distribution for \( i = 1, 2, 3 \), and \( Y_j \) follows \( N_4(0, \Sigma_2) \) distribution for \( j = 1, 2, 3, 4 \), where \( \Sigma_1 \) and \( \Sigma_2 \) are positive definite matrices. Further, let \[ Z = \Sigma_1^{-1/2} X X^T \Sigma_1^{-1/2} + \Sigma_2^{-1/2} Y Y^T \Sigma_2^{-1/2}, \] where \( X = [X_1 \, X_2 \, X_3] \) is a \( 4 \times 3 \) matrix, \( Y = [Y_1 \, Y_2 \, Y_3 \, Y_4] \) is a \( 4 \times 4 \) matrix and \( X^T \) and \( Y^T \) denote transposes of \( X \) and \( Y \), respectively. If \( W_m(n, \Sigma) \) denotes a Wishart distribution of order \( m \) with \( n \) degrees of freedom and variance-covariance matrix \( \Sigma \), 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 \( \{x_1, x_2, \dots, x_n\} \) 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?