Let $N(t)$ be a Poisson process of rate $\lambda$, and $N'(t)$ be a process in which we count only even-numbered arrivals; that is, arrivals 2,4,6... of the Process $N(t)$. Is $N'(t)$ a Poisson process? Hint: sum of independent exponential random variables follows 'Erlang' distribution.
Thanks in advance.
It seems that you are discarding $1,3,5, \ldots$ 'th arrivals and only taking $0, 2, 4,\ldots$'th arrivals. Then the $2$nd arrival from the old process is considered as the $1$st arrival in the new process. The $2$nd arrival time from the old process is the independent sum of exponential random variables, thus it is Erlang 's distribution. The independent sum of exponential pdf can be done by using convolution or by using the moment generating function.