Chapter 6, Problem 6.8STPE

Consider two components and three types of shocks. A type I shock causes component I to fail, a type 2 shock causes component 2 to fail, and a type 3 shock causes both components 1 and 2 to fail. The times until shocks 1, 2, and 3 occur are independent exponential random variables with respective rates ${\lambda }_1$, ${\lambda }_2$ and ${\lambda }_3$. Let $X_i$ denote the time at which component i fails, $i=1,2$. The random variables $X_1,X_2$ are said to have a joint bivariate exponential distribution. Find $P\left\{X_1>s,X_2>t\right\}$.