Three sensors, one silent moment

Question

Three independent detectors register particles as Poisson processes at rates $2$ , $3$ , and $5$ per minute. Their clicks all feed into one shared counter.

What is the probability that the shared counter registers no particle at all during a $12$ -second interval?

Accepted Answer

Because the three detectors are independent Poisson processes, their combined stream is a single Poisson process whose rate is the sum: \lambda = 2 + 3 + 5 = 10 	ext{ per minute}. Now work with that one rate. Twelve seconds is 	frac{12}{60} = 	frac15 of a minute, so the mean number of clicks is \mu = \lambda t = 10 	imes 	frac15 = 2. The probability of zero clicks in the window is just e^{-\mu}: P(N = 0) = e^{-2} \approx 0.135. So about a 13.5\% chance of a silent 12 seconds across all three detectors combined. Takeaway. Superposition extends to any number of independent Poisson sources:…

Three sensors, one silent moment

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