Question:
Let (X, Y) be a random vector having the joint moment generating function
\(M(t_1,t_2)=(\frac{1}{2}e^{-t_1}+\frac{1}{2}e^{t_1})^2(\frac{1}{2}+\frac{1}{2}et_2)^2,\ \ (t_1,t_2)\in \R^2.\)
Then P( |X + Y| = 2) equals __________ (round off to 2 decimal places)
Let (X, Y) be a random vector having the joint moment generating function
\(M(t_1,t_2)=(\frac{1}{2}e^{-t_1}+\frac{1}{2}e^{t_1})^2(\frac{1}{2}+\frac{1}{2}et_2)^2,\ \ (t_1,t_2)\in \R^2.\)
Then P( |X + Y| = 2) equals __________ (round off to 2 decimal places)
\(M(t_1,t_2)=(\frac{1}{2}e^{-t_1}+\frac{1}{2}e^{t_1})^2(\frac{1}{2}+\frac{1}{2}et_2)^2,\ \ (t_1,t_2)\in \R^2.\)
Then P( |X + Y| = 2) equals __________ (round off to 2 decimal places)
Updated On: Oct 1, 2024
Hide Solution
Verified By Collegedunia
Correct Answer: 0.23
Solution and Explanation
The correct answer is 0.23 to 0.27.(approx)
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 \( X \) and \( Y \) be continuous random variables having the joint probability density function
\[ f(x, y) = \begin{cases} e^{-x}, & \text{if } 0 \leq y < x < \infty \\0, & \text{otherwise}\end{cases} \]
Then which of the following statements is/are correct?- IIT JAM MS - 2024
- Statistics
- Multivariate Distributions
- Let \(X_1, X_2, \ldots, X_{10}\) be a random sample from the \(N(3,4)\) distribution, and let \(Y_1, Y_2, \ldots, Y_{15}\) be a random sample from the \(N(-3,6)\) distribution. Assume that the two samples are drawn independently. Define:
\[ \bar{X} = \frac{1}{10} \sum_{i=1}^{10} X_i \]
\[ \bar{Y} = \frac{1}{15} \sum_{j=1}^{15} Y_j \]
\[ S = \sqrt{\frac{1}{9} \sum_{i=1}^{10} (X_i - \bar{X})^2} \]
Then the distribution of \( U = \frac{\sqrt{5}(\bar{X} + \bar{Y})}{S} \) is:- 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
- For \( n \geq 2 \), let \( X_1, X_2, \ldots, X_n \) be a random sample from a distribution with \( E(X_1) = 0 \), \( \text{Var}(X_1) = 1 \), and \( E(X_1^4) < \infty \). Let \[ \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i \] and \[ S_n^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X}_n)^2. \]
Then which of the following statements is/are always correct?- IIT JAM MS - 2024
- Statistics
- Multivariate Distributions
View More Questions
Questions Asked in IIT JAM MS exam
- A bag has 5 blue balls and 15 red balls. Three balls are drawn at random from the bag simultaneously. Then the probability that none of the chosen balls is blue equals
- IIT JAM MS - 2024
- Probability
- Let the random vector \( (X, Y) \) have the joint probability density function
\[f(x, y) = \begin{cases} \frac{1}{x}, & \text{if } 0 < y < x < 1 \\0, & \text{otherwise} \end{cases}.\]
Then \( \text{Cov}(X, Y) \) is equal to:- 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
- 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
- If 12 fair dice are independently rolled, then the probability of obtaining at least two sixes is equal to __________ (round off to 2 decimal places)
- IIT JAM MS - 2024
- Probability
View More Questions