TITLE: Are these facts about the Poisson process correct?
QUESTION [5 upvotes]: Before studying theorems one by one, I want to check whether it is right what I know about Poisson process.
Let $\left\{N(t)\right\}$ be a Poisson process. Then

the number of the event occur during time $t\sim{}Poisson(\lambda{}t)$
Each time interval between adjacent events $\sim{}Exponential(\lambda)$
From any time, time taken until the next event occur $\sim{}Exponential(\lambda)$
Immediately after an event occur, time until $n$ events occur $\sim{}\Gamma(n, \lambda)$
From any time, time taken until $n$ events occur  $\sim{}\Gamma(n, \lambda)$

Are there any wrong sentences? If so, let me know what is wrong. Thank you.

REPLY [4 votes]: Yes, those are correct. Here is more useful information:

The interarrival times are iid's.
The conditional distribution of arrival time $T_1$, $\:P[T_1 \leq \tau \mid N(t)=1]$
with $\tau \leq t$ is uniformly distributed over $(0,t)$, $\:P[T_1 \leq \tau \mid N(t)=1] = \frac{\tau}{t}$. And this generalizes to later times.
A PP has independent increments.
A PP has stationary increments.
A PP is nonstationary, like any process with stationary independent increments.
A PP is a renewal process with exponentially distributed intervals.