# What is the difference between the Poisson distribution and the uniform distribution?

I'm modeling a factory with orders arriving independently of each other. I need a series of actual time stamps of arrivals. I've been told that everyone uses the Poisson distribution to model arrivals. Fine. But what I can't understand is how getting Poisson time stamps differs from getting time stamps with a uniform distribution.

Right now I plan all the arrivals before the simulation starts by choosing uniformly distributed random numbers over the total time span. I sort this list and those are my arrival times. I measure the inter-arrival times and I find that they have an exponential distribution. But I am told that inter-arrival times of Poisson arrival times have an exponential distribution, too.

So what's up? If they have the same distribution of inter-arrival times are Poisson and uniform two names for the same distribution? Is this a discrete vs. continuous distinction?

What is the property of a set of arrival times chosen with Poisson that differs from a set chosen with uniform? Are there fewer small gaps? Fewer big gaps? Something I'm not thinking of?

• For a Poisson process, the times between events follow an exponential distribution. If you pick a time window over which to look at it, the count of events, $n$, follows a Poisson distribution, & given $n$ the time of events follows a uniform distribution. – Scortchi - Reinstate Monica Jun 29 '17 at 16:30