Three persons, A, B and C, get to a post office at the same time. They each want to make a telephone call. There are two telephone booths, which are immediately occupied by A and B. C makes her call after whoever finishes first. They leave the post office as soon as they have completed their calls. We denote by X, Y and Z the length of the telephone calls made by A, B and C, respectively. These three random variables are assumed to be i.i.d., their common law being exponential with parameter λ > 0.
1. Compute the probability that C leaves last.
2. Give the probability distribution of the total time T spent by C in the post office.
3. With 0 being the time of arrival of the three persons at the post office, give the probability distribution of the time of the last departure