Superposition of Two Rate-Specific Poisson Processes

Over any fixed interval, the counts of two independent Poisson processes are independent Poisson variables.

A sum of independent Poisson variables is Poisson with the sum of their means.

1. 1. Sum independent Poisson counts

   Over an interval of length $t$, the two counts are independent Poisson with means $4t$ and $7t$, so their sum is Poisson with mean $(4+7)t$.

2. 2. Read off the combined rate

   The combined process is Poisson with rate equal to the sum of the individual rates.

   $${\lambda }_{\text{total}}=4+7=11$$

Answer: 11