I'm trying to show that a Poisson process with events of two different types can be simulated by 1) creating a Poisson process of events of type 1 and a Poisson process of events of type 2 and adding together 2) one Poisson process can be simulated with 2 event types and split apart.
The expected outcome is that the mean and variance of the sum of the two Poisson processes with only 1 event type in each process should be the same as the mean and variance of the Poisson process with both types of events. This is not the case, the first case's (two separate processes) always has a higher mean and variance than the second.
I'm wondering if there is a mathematical explanation for this and if someone could explain it to me.