Are these facts about the Poisson process correct? 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.

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.

