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Poisson process start with certain assumption about the how process govern in short interval of time $\Delta t$. The first assumption about the Poisson process is that the probability of occurrence of an event in small interval of time $\Delta t$ is given by $\lambda \, \Delta t + o(\Delta t)$, for some constant $\lambda$.

I just wanted to know how this probability (given in above assumption) come from in first place without knowing that process follows Poisson distribution? I found certain derivation on stats.stackexchange but they are based on the assumption that actual process follows Poisson distribution but Poisson distribution is result of the assumptions made in Poisson process.

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    $\begingroup$ This is thoroughly addressed in my answer at stats.stackexchange.com/questions/214421/…. $\endgroup$ – whuber Dec 5 '17 at 18:45
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    $\begingroup$ By "how these probability are derived" do you mean how to derive the fact that the number of events in an interval is Poisson, starting from the assumption in the first paragraph? Or something else? (I don't get what "these probability" refers to) $\endgroup$ – Juho Kokkala Dec 5 '17 at 20:11
  • $\begingroup$ @Juho kokkala I have edited my question. I wanted to know the derivation of probability given in assumption without knowing the fact that the actual process follows Poisson distribution. $\endgroup$ – Neeraj Dec 5 '17 at 20:21
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    $\begingroup$ But what assumptions do you want to start from? (Does @whuber's answer linked above answer your question?) $\endgroup$ – Juho Kokkala Dec 5 '17 at 20:59
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An interesting feature of the Poisson Process is that the Poisson distribution can be derived from properties of the Point Process such as orderliness and complete independence. See e.g. section 1.2 of the book by Cox and Isham Point Processes or in section 2.2 in the book of Daley and Vere Jones An Introduction to the Theory of Point Processes. Roughly speaking, once it is seen from the hypotheses that the interarrivals must be independent and have the same exponential distribution, the Poisson distribution arises.

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  • $\begingroup$ Thanks for your answer. But exponential is also the result of the assumptions made in Poisson process. $\endgroup$ – Neeraj Dec 5 '17 at 20:22
  • $\begingroup$ Even if you never heard of the Poisson distribution before, you can find the exponential interarrival from the simple two hypotheses on the process. You should read the detailed answer cited by whuber who derives it from complete independence and homogeneity. $\endgroup$ – Yves Dec 6 '17 at 6:41

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