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Let's say we have random events happening with the exponential probability: $P(t_i>T)=e^{-\lambda T}$, where $i=1,..,n$. What is the probability that there will be $k$ events within time period $(0,T)$? My understanding is that it should be a Poisson distribution with mean $\lambda$, but why and how can I prove it rigorously?

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    $\begingroup$ Do you mean that the time between events is exponential? The way you have said it right now, it sounds like the time that the $i^{th}$ event occurs is exponentially distributed, which would mean the number of events which occur in $(0,T)$ would have the Binomial distribution $\text{Bin}(n,1-e^{-\lambda T})$, not Poisson. (This is also assuming the times are independent, which you have not specified). Can you clarify? $\endgroup$ Mar 12, 2020 at 14:44
  • $\begingroup$ You are right. Can you write it as an answer? I will accept it. I think the book I am reading is just wrong in claiming it is a Possion distribution. It would become a Poisson if probability is low. $\endgroup$
    – Al Guy
    Mar 15, 2020 at 22:25

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