In a (homogeneous) Poisson process on $\left[0,\infty\right]$ with rate $\lambda$, the waiting time between events is distributed $\text{Exp}\left(\lambda\right)$, i.e. $$p\left(t\right)\propto e^{-\lambda t}$$
For a nonhomogeneous process with intensity $\lambda\left(t\right)$, am I correct in generalizing the waiting time distribution, starting from some time $t_{0}$, as follows? $$p\left(t\mid t_{0}\right)\propto e^{-\int_{t_{0}}^{t}\lambda\left(\tau\right)d\tau}$$