Question:
Let (X, Y) be a discrete random vector. Then which of the following statements is/are true ?
Let (X, Y) be a discrete random vector. Then which of the following statements is/are true ?
Updated On: Oct 1, 2024
- If X and Y are independent, then X2 and |Y| are also independent.
- If the correlation coefficient between X and Y is 1, then P(Y = aX + b) = 1 for some a, b ∈ \(\R\)
- If X and Y are independent and E[(XY)2] = 0, then P(X = 0) = 1 or P(Y = 0) = 1
- If Var(X) = 0, then X and Y are independent
Hide Solution
Verified By Collegedunia
The Correct Option is A, B, C, D
Solution and Explanation
The correct option is (A) : If X and Y are independent, then X2 and |Y| are also independent, (B) : If the correlation coefficient between X and Y is 1, then P(Y = aX + b) = 1 for some a, b ∈ \(\R\), (C) : If X and Y are independent and E[(XY)2] = 0, then P(X = 0) = 1 or P(Y = 0) = 1 and (D) : If Var(X) = 0, then X and Y are independent.
Was this answer helpful?
0
0
Top Questions on Multivariate Distributions
- Suppose that the weights (in kgs) of six months old babies, monitored at a healthcare facility, have \( N(\mu, \sigma^2) \) distribution, where \( \mu \in \mathbb{R} \) and \( \sigma > 0 \) are unknown parameters. Let \( X_1, X_2, \ldots, X_9 \) be a random sample of the weights of such babies. Let \( \overline{X} = \frac{1}{9} \sum_{i=1}^{9} X_i \), \( S = \sqrt{\frac{1}{8} \sum_{i=1}^{9} (X_i - \overline{X})^2} \) and let a 95% confidence interval for \( \mu \) based on \( t \)-distribution be of the form \( (\overline{X} - h(S), \overline{X} + h(S)) \), for an appropriate function \( h \) of random variable \( S \). If the observed values of \( \overline{X} \) and \( S^2 \) are 9 and 9.5, respectively, then the width of the confidence interval is equal to __________ (round off to 2 decimal places) (You may use \( t_{9,0.025} = 2.262, t_{8,0.025} = 2.306, t_{9,0.05} = 1.833, t_{8,0.05} = 1.86 \)).
- IIT JAM MS - 2024
- Statistics
- Multivariate Distributions
- Let \( X \) be a discrete random variable with \( P(X \in \{-5, -3, 0, 3, 5\}) = 1 \). Suppose that \( P(X = -3) = P(X = -5) \), \( P(X = 3) = P(X = 5) \) and \( P(X > 0) = P(X = 0) = P(X < 0) \). Then the value of \( 12P(X = 3) \) is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Statistics
- Multivariate Distributions
- Let \( X \) be a random variable with the moment generating function \[ M(t) = \frac{(1 + 3e^t)^2}{16}, \quad -\infty < t < \infty. \]
Let \( \alpha = E(X) - \text{Var}(X) \). Then the value of \( 8\alpha \) is equal to __________ (answer in integer).- IIT JAM MS - 2024
- Statistics
- Multivariate Distributions
- Let \( X \) be a continuous random variable with a probability density function \( f \) and the moment generating function \( M(t) \). Suppose that \( f(x) = f(-x) \) for all \( x \in \mathbb{R} \) and the moment generating function \( M(t) \) exists for \( t \in (-1, 1) \). Then which of the following statements is/are correct?
- IIT JAM MS - 2024
- Statistics
- Multivariate Distributions
- For \( n \geq 2 \), let \( \epsilon_1, \epsilon_2, \ldots, \epsilon_n \) be i.i.d. random variables having the \( N(0,1) \) distribution. Consider \( n \) independent random variables \( Y_1, Y_2, \ldots, Y_n \) defined by \( Y_i = \beta + \epsilon_i \), \( i = 1,2, \ldots, n \), where \( \beta \in \mathbb{R} \). Define
\[ \bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i \]
\[ T_1 = \frac{2\bar{Y}}{n+1} \]
\[ T_2 = \frac{1}{n} \sum_{i=1}^{n} \frac{Y_i}{i} \]
Then which of the following statements is NOT correct?- IIT JAM MS - 2024
- Statistics
- Multivariate Distributions
View More Questions
Questions Asked in IIT JAM MS exam
- Let \( X_1, X_2, X_3 \) be a random sample from a Poisson distribution with mean \( \lambda \), \( \lambda > 0 \). For testing \( H_0: \lambda = \frac{1}{8} \) against \( H_1: \lambda = 1 \), a test rejects \( H_0 \) if and only if \( X_1 + X_2 + X_3 > 1 \). The power of this test is equal to __________ (round off to 2 decimal places).
- IIT JAM MS - 2024
- Standard Univariate Distributions
- Let \( X_1, X_2, \ldots, X_{10} \) be a random sample from a \( U(-\theta, \theta) \) distribution, where \( \theta \in (0, \infty) \). Let \( X_{(10)} = \max \{ X_1, X_2, \ldots, X_{10} \} \) and \( X_{(1)} = \min \{ X_1, X_2, \ldots, X_{10} \} \). If the observed values of \( X_{(10)} \) and \( X_{(1)} \) are 8 and -10, respectively, then the maximum likelihood estimate of \( \theta \) is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Estimation
- Let \( \Theta \) be a random variable having \( U(0, 2\pi) \) distribution. Let \( X = \cos \Theta \) and \( Y = \sin \Theta \). Let \( \rho \) be the correlation coefficient between \( X \) and \( Y \). Then \( 100\rho \) is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Univariate Distributions
- Consider a coin for which the probability of obtaining head in a single toss is \( \frac{1}{3} \). Sunita tosses the coin once. If head appears, she receives a random amount of \( X \) rupees, where \( X \) has the \( \text{Exp}\left( \frac{1}{9} \right) \) distribution. If tail appears, she loses a random amount of \( Y \) rupees, where \( Y \) has the \( \text{Exp}\left( \frac{1}{3} \right) \) distribution. Her expected gain (in rupees) is equal to __________ (answer in integer).
- IIT JAM MS - 2024
- Probability
- Two factories \( F_1 \) and \( F_2 \) produce cricket bats that are labelled. Any randomly chosen bat produced by factory \( F_1 \) is defective with probability 0.5 and any randomly chosen bat produced by factory \( F_2 \) is defective with probability 0.1. One of the factories is chosen at random, and two bats are randomly purchased from the chosen factory. Let the labels on these purchased bats be \( B_1 \) and \( B_2 \). If \( B_1 \) is found to be defective, then the conditional probability that \( B_2 \) is also defective is equal to __________ (round off to 2 decimal places).
- IIT JAM MS - 2024
- Probability
View More Questions