for an observer arriving at a random time [itex]t_1[/itex], where t=0 is the time when the last car passed, i got the following pdf for [itex]Δ^∗[/itex]- the time the observe waits until the next car:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]ρ_{Δ^∗}=\frac{1}{Δ^∗}⋅(e^{-\frac{Δ^∗}{τ}}−e^{-\frac{2Δ^∗}{τ}})[/itex].

the mean is τ, like the book said and it goes to 0 for [itex]Δ^∗→0[/itex] and [itex]Δ^∗→∞[/itex], but it still looks kind of weird for a probability distribution.... is this correct?

a short summary of the derivation:

[itex]Δ^∗=Δ−t_1[/itex], where Δ is the time between two consecutive cars (as was found in the previous posts).

t1 has a uniform probability distribution between 0 and Δ, therefore:

[itex]ρ_{Δ^∗}=∫ρ_Δ(Δ^∗+ζ)ρ_{t_1}(ζ)dζ[/itex]

for [itex]0<ζ<Δ^∗[/itex]

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# Waiting times - Observer arriving at random time

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