Here is question I am working on to study for an exam that I am quite not sure how to frame a proper answer for:

Trains arrive at a station with i.i.d interarrival times following an exponential distribution with parameter $\lambda$. Suppose you arrive at a station at time t.

(a) Let T0 be the time when the last train arrived before time t. Show t-T0 follows the same exponential distribution as inter-arrival times. (b) Show that the expected inter-arrival time between the last train which arrived before time t and the next train which arrives after time t is $2/\lambda$. Explain why this makes sense.

Attempts : I clearly realize part a) has to with memory-less property of exponential distributions but I am still unable to use that in my proof beyond a proof by argument. As for part b), I am slightly confused by the answer. I assume it uses part a) to split the time up between say T1 and T0 into t-T0 and T1-t which both have exponential distributions? Do not understand the intuition why it is not just the same as the inter-arrival distribution?



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