This is from chapter 6 of the book "Introduction to Probability" written by Dimitri P. Bertsekas.
I am totally lost on how last related property came to be, as there isn't a derivation or a proof of it.
All I could visualize, is a using the definitions. Here is my understanding so far. An interval of length $T+1$, where the amount of arrivals at length $T$ is $N_T$, so the interval from $T$ to 1 has no arrivals. Since $T$ is exponentially distributed, then $T$ has distribution, $\nu e^{-\nu T}$.
After a bit more discussion, I noticed that since, it is finding next arrival after $N_T$. After $T$ it is memory-less so the inter-arrival time of this arrival is counted from $T$ once again, thus explaining the geometric distribution.
However, could someone explain how to derive the parameter of $\frac{\nu}{\lambda+\nu}$ ? Thank you very much