Poisson defects along a wire

Question

Manufacturing defects occur along a wire as a Poisson process at rate $\lambda = 0.5$ per meter.

(a) What is the probability of exactly $1$ defect in a $4$ -meter length? (b) What is the probability that a $3$ -meter stretch is defect-free?

Accepted Answer

A Poisson process works over length (or area, or volume) just as it does over time: the count in an extent L is Poisson with mean \mu = \lambda L. (a) Over 4 meters, \mu = 0.5 	imes 4 = 2: P(N = 1) = e^{-2}\frac{2^1}{1!} = 2e^{-2} \approx 2 	imes 0.1353 = 0.271. (b) Over 3 meters, \mu = 0.5 	imes 3 = 1.5: P(N = 0) = e^{-1.5} \approx 0.223. Takeaway. The Poisson toolkit is not specific to time: set \mu = 	ext{rate} 	imes 	ext{extent} (here defects per meter times meters), then apply P(N = k) = e^{-\mu}\mu^k / k!. Disjoint stretches of wire have independent counts, exactly like disjoint t…

Poisson defects along a wire

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