I am trying to understand the relationship between a Poisson process and the Poisson random variable.
I understand that if $N(t)$ is a Poisson process with rate $\lambda$, then we can consider $N(t)$ ~ Poisson ($\lambda*t$).
Now, I am confused about the various possible random variables that can arise out of the above process, using different values for the rate $\lambda$.
Suppose I have a Poisson process $N(t)$ with rate $1$ (so $\lambda$ $= 1$).
I would like to start with time $0$, and assume some $t$ and $s$ such that $0 < t < s$
I want to understand the relationship between the random variables $N(t)$ and $N(s)$ here.
For example, to calculate $E[N(t) N(s)]$, can I say that $N(s)$ = $N(t)$ + $N(s-t)$ ?
And then I can calculate $E[N(t) N(s)]$ = $E[N(t) * (N(t) + N(s-t))]$
= $E[N(t)^2 + (N(t) N(s-t))]$
= $E[N(t)^2] + E[N(t) N(s-t)]$
Now, $N(t)$ and $N(s-t)$ are independent, since these are the number of events that occur in the disjoint intervals (0, t) and (t, s).
Am I understanding the properties of the Poisson process correctly here? Thank you very much in any assistance with learning this.