Emergency calls from two towns

Question

A dispatch center handles emergency calls from two towns. Town X generates calls as a Poisson process at $3$ per hour and Town Y as an independent Poisson process at $4$ per hour.

What is the probability that the center receives exactly $5$ calls in a given hour?

Accepted Answer

The two towns' call streams are independent Poisson processes, so the center's total call stream is a single Poisson process with the combined rate \lambda = 3 + 4 = 7 	ext{ per hour}. Over one hour the mean count is \mu = \lambda t = 7 	imes 1 = 7. The chance of exactly 5 calls is P(N = 5) = e^{-7}\,\frac{7^5}{5!} = e^{-7}\,\frac{16807}{120} \approx 140.06 	imes 0.000912 \approx 0.128. About a 13\% chance. There was no need to consider the many ways the five calls could split between the two towns; merging handles all of that automatically. Takeaway. Merging independent Poisson processes …

Emergency calls from two towns

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