Property of Poisson Process

The Problem

Customers arrive at a store according to a Poisson process with rate $\lambda$ per hour.

1. Given that exactly $n$ customers arrived in the time interval $[0,T]$, what is the joint distribution of their arrival times?
2. Show that the inter-arrival times are i.i.d. $\text{Exponential}(\lambda )$.
3. Memoryless property: Given that no customer has arrived in the last $t$ hours, what is the distribution of the additional time until the next arrival?
4. Order statistics property: Given $n$ arrivals in $[0,T]$, the arrival times (sorted) have the same distribution as the order statistics of $n$ i.i.d. $\text{Uniform}(0,T)$ random variables. Prove this.