An electronic system consists of two components arranged in series; that is, the system works if and only if both of the components are working. Each component operates independently and has a random lifetime (i.e., time to failure) that follows an exponential distribution, with parameter 1 for component 1 and 2 for component 2. Let X1 be the lifetime of component 1 and X2 be lifetime of component 2.
a. Find the distribution function for the system lifetime T (hint: write an expression for the event that the system has failed by time t in terms of X1 and X2).
b. What is the probability that component 1 fails before component 2?