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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.

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Let $A$ and $B$ be the waiting times.

Let's guess that these are exponentially distributed random variables with $P(A \geq t) = P(B \geq t) = e^{-\frac12 \lambda t}$.

By this answer, we have for the r.v. $C := \min\{A,B\}$ that $P(C \geq t) = e^{-\lambda t}$. This is consistent with the assumption (if I understand it correctly).

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