Combined clicks from two Geiger counters

Question

Two independent Geiger counters register clicks as Poisson streams: counter A averages 7 clicks per minute and counter B averages 11 clicks per minute. Let $C$ be the combined number of clicks per minute.

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

Accepted Answer

Independent Poisson streams superpose into a single Poisson whose rate is the sum of the two rates: C \sim 	ext{Poisson}(7 + 11) = 	ext{Poisson}(18), \qquad \mathbb{E}[C] = 18, \qquad \operatorname{Var}(C) = 18. Standard deviation \sqrt{18} \approx 4.24. As always for a Poisson, the mean and the variance are the same number. Takeaway. Two independent counters pool to Poisson(18); mean equals variance at 18.

Combined clicks from two Geiger counters

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