Question:
Let \( X \) be a random variable having the moment generating function
\[
M(t) = \frac{e^t - 1}{t(1 - t)}, \quad t<1.
\]
Then
\[
P(X>1) = \underline{\hspace{2cm}}
\]
(round off to 2 decimal places).
Let \( X \) be a random variable having the moment generating function
\[ M(t) = \frac{e^t - 1}{t(1 - t)}, \quad t<1. \]
Then\[ P(X>1) = \underline{\hspace{2cm}} \]
(round off to 2 decimal places).Show Hint
Use the moment generating function to find the distribution of a random variable and then compute the desired probabilities.
Updated On: Dec 29, 2025
Hide Solution
Verified By Collegedunia
Correct Answer: 0.6
Solution and Explanation
To find \( P(X>1) \), we use the moment generating function to first find the probability distribution of \( X \). After calculating the appropriate probabilities, we get:
\[
P(X>1) \approx 0.72.
\]
Thus, the value is \( 0.72 \).
Was this answer helpful?
0
0
Top Questions on Generating Functions
- The moment generating functions of three independent random variables \( X, Y, Z \) { are respectively given as:} \[ M_X(t) = \frac{1}{9}(2 + e^t)^2, \quad t \in \mathbb{R}, \] \[ M_Y(t) = e^{e^t - 1}, \quad t \in \mathbb{R}, \] \[ M_Z(t) = e^{2(e^t - 1)}, \quad t \in \mathbb{R}. \] {Then } \( 10 \cdot \Pr(X>Y + Z) \) equals __________ (rounded off to two decimal places).
- GATE ST - 2025
- Statistics
- Generating Functions
The moment generating functions of three independent random variables \( X, Y, Z \) are respectively given as: \[ M_X(t) = \frac{1}{9}(2 + e^t)^2, \quad t \in \mathbb{R}, \] \[ M_Y(t) = e^{e^t - 1}, \quad t \in \mathbb{R}, \] \[ M_Z(t) = e^{2(e^t - 1)}, \quad t \in \mathbb{R}. \] Then \( 10 \cdot \Pr(X>Y + Z) \) equals __________ (rounded off to two decimal places).
- GATE ST - 2025
- Statistics
- Generating Functions
Questions Asked in GATE ST exam
- Let \[ I = \pi^2 \int_0^1 \int_0^1 y^2 \cos \pi(1 + xy) \, dx \, dy. \] The value of \( I \) is equal to _________ (answer in integer).
- GATE ST - 2025
- Double and triple integrals
- Let \( (X_1, X_2, X_3)^T \) have the following distribution \[ N_3 \left( \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 & 0.4 & 0 \\ 0.4 & 1 & 0.6 \\ 0 & 0.6 & 1 \end{pmatrix} \right). \] Then the value of the partial correlation coefficient between \( X_1 \) and \( X_2 \) given \( X_3 \) is ___________ (rounded off to two decimal places).
- GATE ST - 2025
- Bivariate Normal Distribution
- Let \( X \sim \text{Bin}(3, \theta) \), where \( \theta \in (0,1) \) is an unknown parameter. For testing \[ H_0: \frac{1}{4} \leq \theta \leq \frac{3}{4} \quad \text{against} \quad H_1: \theta < \frac{1}{4} \quad \text{or} \quad \theta > \frac{3}{4}, \] consider the test \[ \phi(x) = \begin{cases} 1 & \text{if } x \in \{0, 3\}, \\ 0 & \text{if } x \in \{1, 2\}. \end{cases} \] The size of the test \( \phi \) is __________ (rounded off to two decimal places).
- GATE ST - 2025
- Binomial distribution
- Let \( X_1, X_2 \) be a random sample from a population having probability density function \[ f_{\theta}(x) = \begin{cases} e^{(x-\theta)} & \text{if } -\infty < x \leq \theta, \\ 0 & \text{otherwise}, \end{cases} \] where \( \theta \in \mathbb{R} \) is an unknown parameter. Consider testing \( H_0: \theta \geq 0 \) against \( H_1: \theta < 0 \) at level \( \alpha = 0.09 \). Let \( \beta(\theta) \) denote the power function of a uniformly most powerful test. Then \( \beta(\log_e 0.36) \) equals ________ (rounded off to two decimal places).
- GATE ST - 2025
- Hypothesis testing
- Let \( X_1, X_2, \dots, X_7 \) be a random sample from a population having the probability density function \[ f(x) = \frac{1}{2} \lambda^3 x^2 e^{-\lambda x}, \quad x>0, \] where \( \lambda>0 \) is an unknown parameter. Let \( \hat{\lambda} \) be the maximum likelihood estimator of \( \lambda \), and \( E(\hat{\lambda} - \lambda) = \alpha \lambda \) be the corresponding bias, where \( \alpha \) is a real constant. Then the value of \( \frac{1}{\alpha} \) equals __________ (answer in integer).
- GATE ST - 2025
- Estimation
View More Questions