List of top Statistics Questions on Exponential distribution

The service times (in minutes) at two petrol pumps \( P_1 \) and \( P_2 \) follow distributions with probability density functions \[ f_1(x) = \lambda e^{-\lambda x}, \quad x>0 \quad {and} \quad f_2(x) = \lambda^2 x e^{-\lambda x}, \quad x>0, \] respectively, where \( \lambda>0 \). For service, a customer chooses \( P_1 \) or \( P_2 \) randomly with equal probability. Suppose, the probability that the service time for the customer is more than one minute, is \( 2e^{-2} \). Then the value of \( \lambda \) equals __________ (answer in integer).

The service times (in minutes) at two petrol pumps \( P_1 \) and \( P_2 \) follow distributions with probability density functions \[ f_1(x) = \lambda e^{-\lambda x}, \quad x>0 \quad {and} \quad f_2(x) = \lambda^2 x e^{-\lambda x}, \quad x>0, \] respectively, where \( \lambda>0 \). For service, a customer chooses \( P_1 \) or \( P_2 \) randomly with equal probability. Suppose, the probability that the service time for the customer is more than one minute, is \( 2e^{-2} \). Then the value of \( \lambda \) equals _________ (answer in integer).

Let \( X, Y_1, Y_2 \) be independent random variables such that \( X \) has the probability density function \[ f(x) = \begin{cases} 2e^{-2x} & \text{if } x \geq 0, \\ 0 & \text{otherwise}, \end{cases} \] and \( Y_1 \) and \( Y_2 \) are identically distributed with probability density function \[ g(x) = \begin{cases} e^{-x} & \text{if } x \geq 0, \\ 0 & \text{otherwise}. \end{cases} \] For \( i = 1, 2 \), let \( R_i \) denote the rank of \( Y_i \) among \( X, Y_1, Y_2 \). Then \( E(R_1 + R_2) \) equals:

Let \( X \) be a random variable with pdf \[ f(x) = \begin{cases} e^{-x}, & x > 0 \\ 0, & x \le 0 \end{cases} \] Define \( Y = \lfloor X \rfloor \), the greatest integer less than or equal to \( X \). Then \(E(Y^2)\) is equal to:

Let \[ \alpha = \lim_{n \to \infty} \left(1 + n \sin \frac{3}{n^2}\right)^{2n}. \] Then, \(\ln \alpha\) is equal to ................

Let \( B \) denote the length of the curve \( y = \ln(\sec x) \) from \( x = 0 \) to \( x = \frac{\pi}{4} \). Then, the value of \( 3\sqrt{2}(e^B - 1) \) is equal to .............

Let \(X_1, X_2, ..., X_n\) be a random sample from an exponential distribution with probability density function \[ f(x; \theta) = \begin{cases} \theta e^{-\theta x}, & x>0,
0, & \text{otherwise,} \end{cases} \] where \(\theta \in (0, \infty)\) is unknown. Let \(\alpha \in (0,1)\) be fixed and let \(\beta\) be the power of the most powerful test of size \(\alpha\) for testing \(H_0: \theta = 1\) against \(H_1: \theta = 2\). Consider the critical region \[ R = \left\{ (x_1, x_2, ..., x_n) \in \mathbb{R}^n : \sum_{i=1}^n x_i>\frac{1}{2}\chi^2_{2n}(1-\alpha) \right\}, \] where for any \(\gamma \in (0,1)\), \(\chi^2_{2n}(\gamma)\) is a fixed point such that \( P(\chi^2_{2n}>\chi^2_{2n}(\gamma)) = \gamma. \) Then, the critical region \(R\) corresponds to the

Let \( X_1, X_2, X_3 \) be a random sample from a distribution with the probability mass function

If \( \hat{X}(X_1, X_2, X_3) \) is an unbiased estimator of \( \theta \), which of the following CANNOT be attained as a value of the variance of \( \hat{X} \) at \( \theta = 1 \)?