Poisson process properties proof

Question

i started to study about poisson process and i having a problem with the next question:

M(t) is poisson process with with parameter -x.
Ti is the first time that M(Ti)=i.

prove that (Ti+2 - Ti) has a Erlang Distribution with parameter x,k=2.

i understand why the distribution of (Ti+1 - Ti) has Exp Distribution , but im not sure what do to when i have an arrival in my time interval. hope it was clear. thanks.

Answer 1

$T_{i+2}-T_i\stackrel{d}{=}T_2$

If $\lambda$ denotes the parameter of the Poisson process then:$$P(T_{i+2}-T_i>t)=P(T_2>t)=P(N_t\leq1)=e^{-\lambda t}\left[\frac{(\lambda t)^0}{0!}+\frac{(\lambda t)^1}{1!}\right]=e^{-\lambda t}[1+\lambda t]$$

Answer 2

HINT

$T_{i+2}-T_i = (T_{i+2}-T_{i+1})+(T_{i+1}-T_i)$