There are two bus lines (A & B) that stop at a bus station. The time until the next bus arrives is distributed as an exponential random variable with parameter λ buses/hour for both bus lines together. It is equally likely for a bus to arrive from line A and line B. What is the distribution for the waiting time for a bus from each line to arrive?
Attempt: I understand that after the first bus arrives, the number of arrivals it takes for a bus from the other line to come is distributed as a geometric RV (let's call it X). I think the distribution of the waiting time for a bus from each line to arrive would be given by the sum of the exponential RV over X. I'm not sure where to go from here to calculate the distribution.