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We know that when the arriving rate of a Poisson process $X(t)$ becomes constant, then the process becomes a homogeneous Poisson process.

I have trouble understanding what "a constant arriving rate" means. My confusion largely arises from the fact that any ensemble $X(t_0)$ ($t_0$ is a constant) is in fact a random variable, implying the number of points arriving between $[0, t_0]$ is random.

Given that I can't even say for sure how many points arrive between $[0, t]$, how can I say I have "a constant arriving rate"?

In real life, what is an example case where we have a constant arriving rate, but we can't say for sure how many points will arrive for a known period of time? It just sounds contradictory to me.

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Consider an analogous situation. Imagine you're tossing a completely fair coin. In a thousand tosses, the chance that you get 500 heads is about 2.5%, so almost every time you tried, the sample proportion of heads would differ from 50%.

We have to be careful to keep population quantities in our models conceptually separate from their sample equivalents.

"Constant arrival rate" is a statement about expected (i.e. population) quantities in a Poisson process model. If the model is correct, the actual observations will obviously still show variation about the expected arrival rate.

One example of a phenomenon that seems to be suitable for modelling as a Poisson process is the number of fast radio bursts over time (at least over short periods -- if we looked across many millions of years, the rate wouldn't be constant).

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If the Poisson process has settled on a constant rate $\lambda$ then you can conceptualize it like this:

  • In the next $k$ time units you expect, on average, $k\lambda$ events to occur. This doesn't mean that this many events actually will occur ($k\lambda$ may not even be a whole number!) but this is the distinction between population and sample that @Glen_B was emphasising. Your (population) model for the number of events in the next $k$ time units is a Poisson distribution with parameter $k\lambda$. You could count the number of events in the next $k$ time units, then in the next $k$, then in the next, until you have a sample of the number of events occurring in $n$ time intervals. If your model of homogeneity is correct and $n$ is sufficiently large then the relative frequencies of no events, one event, two events etc will settle down close to the probabilities $\mathsf{P}(X=0)$, $\mathsf{P}(X=1)$, $\mathsf{P}(X=2)$ etc calculated from a $\text{Poisson}(k\lambda)$ distribution, and the mean count in an interval will be near to the $k\lambda$ you expected.
  • The expected time until the next event is $\lambda^{-1}$ and your model for the population those times are drawn from is exponential with parameter $\lambda$. If you sample a large number of events then you will find their inter-arrival times will have a mean value close to $\lambda^{-1}$ and their histogram will look very similar to the PDF of the theoretical $\text{Exp}(\lambda)$ distribution.

(You can read "time units" as "space units" for a spatial process.)

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