List of top Statistics Questions on Exponential distribution asked in GATE ST

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: