Chapter 5.2, Problem 13E

Let $X_1$, $X_2$ be independent random variables representing lifetimes (in hours) of two key components of a device that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\min \left(X_1,X_2\right)$ and $Y_2=\max \left(X_1,X_2\right)$, so that the space of $Y_1,Y_2$ is $0<y_1<y_2<\infty$

(a) Find $G\left(y_1,y_2\right)P\left(Y_1\le y_1,Y_2\le y_2\right)$.

(b) Compute the probability that the device fails after 1200 hours: that is, compute $P\left(Y_2>1200\right)$.