Are these facts about the Poisson process correct?

Question

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.

But in general you cannot be sure that the process $N(t)$ is a homogeneous Poisson process. — J. Goles, Oct 26 '18 at 09:29

score 4 · Accepted Answer · answered May 09 '16 at 13:49

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.

I'm impressed with your nice answer. Thank you. – Danny_Kim May 09 '16 at 14:05 — Danny_Kim, May 09 '16 at 14:05

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