Let the density probability function of the length of intervals of a renewal process be $ p(x)=\rho e^{-\rho(x-\gamma)}, x>\gamma>0$. Show that if $N_t$ is the number of events occured in the interval $(0,t)$ then
$P{}(N_t<r)=\sum_{k=0}^{r-1} e^{-\rho(t-r\gamma)}\frac{(\rho(t-r\gamma))^{k}}{k!}$
My idea was to prove that the random variables $N_t$ have Poisson distribution and so the given probability will be disjoint union of $P(N_t=0), P(N_t=1),...,P(N_t=r-1)$ and so we will have $P{}(N_t<r)=\sum_{k=0}^{r-1} P(N_t=k)$. But I dont know if this is the right way, or should i start from somwhere else.
Thank you for your help!