Total contacts at a help desk (sum of Poissons)

Question

A help desk receives phone calls as a Poisson stream averaging 6 per hour and, independently, chat messages as a Poisson stream averaging 10 per hour. Let $T$ be the total number of contacts (calls plus chats) in one hour.

What distribution does $T$ follow, and what are its mean and variance?

Accepted Answer

Two facts about Poisson counts: 1. A Poisson count has the unusual property that its mean and its variance are the same number, \lambda. 2. Superposition: if you add two independent Poisson streams with rates \lambda1 and \lambda2, the combined stream is again Poisson with rate \lambda1 + \lambda2. Random arrivals from separate independent sources simply pool. So T \sim 	ext{Poisson}(6 + 10) = 	ext{Poisson}(16). Therefore \mathbb{E}[T] = 16, \qquad \operatorname{Var}(T) = 16. Standard deviation \sqrt{16} = 4. Takeaway. Independent Poisson streams add: rates sum to 16, and for any Poisson th…

Total contacts at a help desk (sum of Poissons)

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