Merging two Poisson streams

Question

Order flow into a venue comes from two independent sources: retail orders arrive as Poisson $(\lambda_1)$ per minute and institutional orders as Poisson $(\lambda_2)$ per minute.

What is the distribution of the total order count per minute, and, conversely, if you randomly label each order, what governs the sub-streams?

Accepted Answer

Merging: the sum is Poisson. The sum of independent Poisson variables is Poisson, with the rates adding: N1 + N2 \sim 	ext{Poisson}(\lambda1 + \lambda2). A quick proof uses moment generating functions. A Poisson(\lambda) has MGF \exp\{\lambda(e^t - 1)\}. By independence, the MGF of the sum is the product: \exp\{\lambda1(e^t-1)\}\cdot\exp\{\lambda2(e^t-1)\} = \exp\{(\lambda1+\lambda2)(e^t - 1)\}, which is exactly the MGF of a Poisson(\lambda1+\lambda2). Since the MGF determines the distribution, the total is Poisson with the combined rate. This extends to any number of independent Poisson str…

Merging two Poisson streams

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