# Derivation of probability under assumption of Poisson process

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.

• This is thoroughly addressed in my answer at stats.stackexchange.com/questions/214421/…. – whuber Dec 5 '17 at 18:45
• 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) – Juho Kokkala Dec 5 '17 at 20:11
• @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. – Neeraj Dec 5 '17 at 20:21
• But what assumptions do you want to start from? (Does @whuber's answer linked above answer your question?) – Juho Kokkala Dec 5 '17 at 20:59