Question:
Let (π, π, π) be a random vector having the joint probability density function
\(f(x,y, z) =\begin{cases} \frac{1}{2\,xy}, & \quad \text{if }0<z<y<x<1.\\ \frac{1}{2x^2}, & \quad if\,0<z<x<y<2x<2,\\0, & \quad\,\,Otherwise.\end{cases}\)
Then, which one of the following statements is FALSE?
Let (π, π, π) be a random vector having the joint probability density function
\(f(x,y, z) =\begin{cases} \frac{1}{2\,xy}, & \quad \text{if }0<z<y<x<1.\\ \frac{1}{2x^2}, & \quad if\,0<z<x<y<2x<2,\\0, & \quad\,\,Otherwise.\end{cases}\)
Then, which one of the following statements is FALSE?
\(f(x,y, z) =\begin{cases} \frac{1}{2\,xy}, & \quad \text{if }0<z<y<x<1.\\ \frac{1}{2x^2}, & \quad if\,0<z<x<y<2x<2,\\0, & \quad\,\,Otherwise.\end{cases}\)
Then, which one of the following statements is FALSE?
Updated On: Oct 1, 2024
- \(π(π < π < π) = \frac{1}{2}\)
- π(π < π < π) = 0
- πΈ(min{π, π}) =\(\frac{1}{4}\)
- πππ (π | π = \(\frac{1}{2}\) ) = \(\frac{1}{12}\)
Hide Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
The correct option is (C): πΈ(min{π, π}) =\(\frac{1}{4}\)
Was this answer helpful?
0
0
Top Questions on Univariate Distributions
- Suppose that the random variable \( X \) has an \( \text{Exp}\left(\frac{1}{5}\right) \) distribution, and for any \( x > 0 \), the conditional distribution of the random variable \( Y \), given \( X = x \), is \( N(x, 2) \). Then \( \text{Var}(X + Y) \) is equal to:
- IIT JAM MS - 2024
- Statistics
- Univariate Distributions
- For \( n \in \mathbb{N} \), let \( X_1, X_2, \ldots, X_n \) be a random sample from the Cauchy distribution having probability density function
\[f(x) = \frac{1}{\pi(1 + x^2)}, \quad -\infty < x < \infty.\]
Let \( g: \mathbb{R} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} x, & \text{if } -1000 \leq x \leq 1000 \\0, & \text{otherwise}\end{cases}.\]
Let
\[\alpha = \lim_{n \to \infty} P\left( \frac{1}{n^{3/4}} \sum_{i=1}^n g(X_i) > \frac{1}{2} \right).\]
Then \( 100\alpha \) is equal to __________ (answer in integer).- IIT JAM MS - 2024
- Statistics
- Univariate Distributions
- Let \( f_1(x) \) be the probability density function of the \( N(0,1) \) distribution, and \( f_2(x) \) be the probability density function of the \( N(0, 6) \) distribution. Let \( Y \) be a random variable with probability density function
\[f(x) = 0.6 f_1(x) + 0.4 f_2(x), \quad -\infty < x < \infty.\]
Then \( \text{Var}(Y) \) is equal to:- IIT JAM MS - 2024
- Statistics
- Univariate Distributions
- Let π be a continuous random variable having the π(β2, 3) distribution. Then which of the following statements is correct?
- IIT JAM MS - 2024
- Statistics
- Univariate Distributions
- Let \( X_1, X_2, X_3 \) be i.i.d. random variables from a continuous distribution having probability density function
\[f(x) = \begin{cases} \frac{1}{2}x^{-3}, & \text{if } x > \frac{1}{2} \\0, & \text{if } x \leq \frac{1}{2} \end{cases}.\]
Let \( X_{(1)} = \min\{X_1, X_2, X_3\} \). Then the value of \( 10E(X_{(1)}) \) is equal to __________ (answer in integer).- IIT JAM MS - 2024
- Statistics
- Univariate 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