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Suppose $N_t$ is poisson with parameter $\lambda t$. Suppose further that $T_1$ is the random variable that represents the time until the first arrival.

I understand very well that the event $\{N_t = 0 \}$ corresponds to the event $\{T_1 > t \}$ because of course saying that I had no arrivals unitl $t$ implies that the time until the first arrival is greater than $t$. From this relationship I can easily derive the exponential distribution.

My question is: shouldn't the event $\{N_t = 1 \}$ correspond to the event $\{T_1 \leq t \}$? Because saying that in the interval $[0,t]$ we had exactly one arrival is the same as saying that the time until the first arrival is less than or equal to $t$. But then I cannot derive the exponential from this. Where is the catch?

gioxc88
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    The catch is that $N_t>0$ corresponds to $T_1\le t$ – Aksakal Mar 12 '18 at 18:06
  • I understand this with formulas, simply because of $N_t>0$ is the complement of the sample space, but I miss the intuition. Could you be so kind to make an argument of why my reasoning is wrong? It seems so flawless to me, yet I know it is wrong. – gioxc88 Mar 12 '18 at 20:52
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    Consider $N_t=2$, it means that $T_10$, then $T_2>0$ too. – Aksakal Mar 12 '18 at 20:56
  • ok I understand that now .. what a stupid question. So the event $\{N_t = 1 \}$ is contained in the event $\{T_1 \leq t \}$ because $T_1 \leq t$ also if $N_t =1,2,...$ thank you – gioxc88 Mar 12 '18 at 21:03
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    Perhaps reviewing the derivation of the relationship at https://stats.stackexchange.com/questions/214421 will reveal the key ideas and assumptions, helping you solidify the concepts. – whuber Mar 12 '18 at 21:56

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