This is the way I introduce this:
- We're talking about a Poisson process.
- Events happen at rate $\lambda(t)$.
- The number of events that happen between times $a$ and $b$ is Poisson-distributed with mean $\int_a^b \lambda(t) ,dt$. That's what "Poisson process" means.
- Given a number of events that happen, the time of those events is distributed with pdf $\lambda(t) / \left(\int_a^b \lambda(s) ,ds\right)$.
And, from one point of view, that's all there is to know about inhomogeneous Point processes!
But now say you want to start at time $a$ and ask when the next event is? It turns out that math is a lot harder. Then you descend into all the other stuff.
cc @dinacmistry @RobertJacobsonCDC -- I was in a sync convo w/ Dina about this
This is the way I introduce this:
And, from one point of view, that's all there is to know about inhomogeneous Point processes!
But now say you want to start at time$a$ and ask when the next event is? It turns out that math is a lot harder. Then you descend into all the other stuff.
cc @dinacmistry @RobertJacobsonCDC -- I was in a sync convo w/ Dina about this