Poisson process: nth jump from expectation of interarrival time

Question

The interarrival time for a poisson process is given as

$ \Bbb E[T_i] = 1/\lambda $

How can I compute the arrival time for the $nth$ jump from this. Surely its not equal to $n/\lambda$ ?

Answer 1

I would say that the time of occurrence 'n' events in the Poisson Process (ie the sum of n values ​​of the random variables with exponential probability distribution) has a Erlang probability distribution (special case of gamma probability distribution).

The mean value $E(t) = \frac{n}{\lambda}$ and dispersion $D = \frac{n}{\lambda^2}$