# Why does the unordered arrivals in Poisson process iid uniform when conditioned by $N(t)=k$?

Let $$N(t)$$ be the number of arrivals in the Poisson process of rate $$\lambda$$.

I already know that the 'ordered' arrival times are uniformly distributed on the region $$0 under $$N(t)=k$$, but I'm not sure that I can recover the unordered distribution from ordered distribution. I'm even not sure that the unordered ones are independent.

How should I approach for the distributions of unordered arrivals?

• It does not make sense to talk about the unordered ones being independent or not; you cannot speak of $X_1$ being independent of $X_2$, since the arrivals are not numbered. Can you be more specific about what you want to know about the unordered distribution? It seems that it is full described by saying it is uniform over the set $\{(t_1,\dots,t_k)\mid 0<t_1<t_2<\dots<t_k<t\}\subseteq \mathbb R^k$. – Mike Earnest Apr 16 at 15:14