Consider a Poisson process. Given that a single arrival occurred in a given interval [0,t], why is the resulting distribution for the arrival time uniform?
2026-03-28 01:13:37.1774660417
Why arrival time in Poisson process has uniform distribution given that a single arrival occurred in an interval
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Let the arrival times be $\sim \exp(\alpha).$ Denote $S_1$ to be the time of the first arrival,and $N_t $ to be the number of arrivals upto time $t$. $$P(S_1\le x\ |\ N_t=1)={P(S_1\le x,N_t=1)\over P(N_t=1)}\\={P(N_x= 1,N_t-N_x=0)\over P(N_t=1)}\\={P(N_x= 1)P(N_t-N_x=0)\over P(N_t=1)}\\={P(N_x= 1)P(N_{t-x}=0)\over P(N_t=1)}\\={\alpha xe^{-\alpha x}\cdot e^{-\alpha (t-x)}\over \alpha te^{-\alpha t}} ={x\over t} $$