# Cumulative distribution function of the sum of waiting times

Suppose that a professor schedules a meeting with two students, with meeting time (in hours) modeled as iid random variables, each exponentially distributed with parameter 3. The first student is on time, but the second student is 5 minutes late. If the first student has not finished his meeting by the time the second student arrives, the second student waits for the first to finish. What is the cumulative distribution function of the random variable $$R$$, which is the time between the arrival of the first student and the departure of the second student?

I've tried using the law of total probability and a maximum function, but I can't get either to work out. Any tips? Note that this is homework, so please don't totally solve the problem.

• You want the distribution of $T=\max\{T_1,t_a\}+T_2$, where $T_i$ are the IID length of meeting times with the professor for each student $i$ and $t_a$ is the arrival time of student $2$. Is this what you tried to compute the CDF of? Where did it go wrong? Here’s one hint, conditional on $T_1>t_a$, $T$ is the sum of IID exponential RVs hence is Gamma distributed. Conditional on $T_1\leq t_a$ then $T$ distributed like a shifted exponential RV, which has CDF... and so unconditionally... – Nap D. Lover Oct 26 '18 at 1:15