題組內容

8. (10%) Two components operate in parallel in a device. The device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with the probability density function. The cost of operating the device is $Z = Y_1 + 2Y_2$, where $Y_1 = \min(X_1, X_2)$ and $Y_2 = \max(X_1, X_2)$

(a) Find the joint probability density function of \( Y_{1} \) and \( Y_{2} \).