Question:
Let π and π be jointly distributed random variables such that, for every fixed π>0, the conditional distribution of π given π=π is the Poisson distribution with mean π. If the distribution of π is πΊππππ (2, \(\frac{1}{2}\) ), then the value of π(π= 0)+π(π=1) equals
Let π and π be jointly distributed random variables such that, for every fixed π>0, the conditional distribution of π given π=π is the Poisson distribution with mean π. If the distribution of π is πΊππππ (2, \(\frac{1}{2}\) ), then the value of π(π= 0)+π(π=1) equals
Updated On: Oct 1, 2024
- \(\frac{7}{27}\)
- \(\frac{20}{27}\)
- \(\frac{8}{27}\)
- \(\frac{16}{27}\)
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
The correct option is (B): \(\frac{20}{27}\)
Was this answer helpful?
0
0
Top Questions on Standard Univariate Distributions
- 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
- Statistics
- Standard Univariate Distributions
- Let \( X \) be a random variable having the Poisson distribution with mean \( 1 \). Let \( g: \mathbb{N} \cup \{0\} \to \mathbb{R} \) be defined by
\[g(x) = \begin{cases} 1 & \text{if } x \in \{0, 2\} \\ 0 & \text{if } x \notin \{0, 2\} \end{cases}\]
Then \( E(g(X)) \) is equal to:- IIT JAM MS - 2024
- Statistics
- Standard Univariate Distributions
Questions Asked in IIT JAM MS exam
- 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
- Multivariate 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
- 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
- Multivariate Distributions
View More Questions