Superposition of Two Poisson Processes

Think about the arrival counts over any fixed interval and how independent Poisson counts combine.

Recall that a sum of independent Poisson random variables is itself Poisson with the summed parameter.

1. 1. Combine counts over an interval

   Over any interval of length $t$, the counts of the two processes are independent Poisson random variables with means ${\lambda }_{1}t$ and ${\lambda }_{2}t$.

2. 2. Apply the sum of independent Poissons

   Since a sum of independent Poisson random variables is Poisson with the sum of their means, the combined process is itself Poisson.

   $$N_1(t)+N_2(t)\sim \text{Poisson}(({\lambda }_1+{\lambda }_2)t)$$

Answer: Poisson process with rate lambda_1+lambda_2