Poisson Distribution interarrivial time

Question

It is obvious that given that in the interval $(0,t)$ the number of arrivals following Poisson process is $N(t)=n$, these $n$ arrivals are independently and uniformly distributed in the interval. It is also clear that the interarrival time of the events following poisson process follows the exponential distribution. Here is my point. I think If $n$ arrivals follows uniform distribution, the interarrivail time of them should follow uniform distribution as well. Am I doing wrong? Anybody have an Idea, pls explain it to me.

Answer 1

The joint distribution of inter-arrival times is uniform on $\{x_1+\dots + x_n<t\}$ conditionally on the fact that $N(t) = n$. Unconditionally, they are independent and have exponential distribution. There is no contradiction; you can try to verify this fact yourself, it's not that hard.

Moreover, as @Did wrote, while the joint (conditional) distribution of the inter-arrival times is uniform, the individual (conditional) distribution of each inter-arrival time is not.

The reason of this non-uniformity comes out of the fact that, conditionally on $N(t) = n$, the inter-arrival times are no longer independent. If they were, then, naturally, you would be able to deduce the uniformity of marginal distributions from that of their joint distribution.

Another way to put this. The conditional distribution of the arrival times is uniform on $[0,t]^n$, so indeed they can be understood as iid uniform random variables on $[0,t]$. However, if you are interested in the inter-arrival times, then you should first arrange the arrival times in the increasing order. And at this point you already lose independence of individual components.