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I am studying a scenario where (N) particles are emitted at time (t=0), and the detection time since the origin follows an exponential distribution with time constant (L). Specifically, the PDF is given by:

P(T)=1/L * exp(-T/L)

In my simulations, I observed that the time distribution between two consecutive detections is also exponential, with time constant (L/(N-1)). I am trying to understand and derive this relationship.

Can someone please provide a hint as to where I can get the derivation. Thank you for your insights.

Vova N
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    See https://math.stackexchange.com/a/80485/631742, conditionally on the $k$-th order statistic, the $k+1$-st order statistic of i.i.d. exponential random variables behaves like the $k$-th order statistic plus another independent exponential random variable, as you seem to observe numerically. – Maximilian Janisch Nov 25 '23 at 21:34

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