What is the distribution similar to the Poisson distribution with non-constant mean rate?

Question

As the Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.
I have been asked the question that what is the distribution if the mean rate is non-constant?
I thought in this case it is about the sum of Poisson random variables which is also a Poisson random variable with a mean rate equal to the sum of the mean rates of each random variable.
Is it correct?

Answer 1

If you have a Poisson process with constant rate $\lambda$, then the distribution of the number of events from time $t_1$ to time $t_2$ has a Poisson distribution with mean $(t_2-t_1)\lambda$

If instead the rate varies as a function of time so $\lambda(t)$ with events occurring independently of the time since the last event (an inhomogeneous Poisson process), then the distribution of the number of events from time $t_1$ to time $t_2$ has a Poisson distribution with mean $\int\limits_{t_1}^{t_2} \lambda(t)\, dt$