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Let's say $N(t)$ is a poisson process with parameter $\lambda$. Let $W_n$ to represent the waiting time of $n$th arrival. What is $P(W_{N(t)+2}≤t+s)$? I solved this question by the following way:$$P(W_{N(t)+2}≤t+s)=P(N(t,t+s] ≥2)$$$$=1-e^{\lambda s}-\lambda s e^{\lambda s}$$ so basically $W_{N(t)+2}$ has the same distribution as $W_2$ and the only difference is the time shifted to $t$. Is my proof right? Is this the rigorous way to reach the answer? It looks so straightforward.

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