Two meteors in half an hour

Question

Meteors streak across the sky during a shower as a Poisson process at rate \lambda = 5 per hour. What is the probability that you see exactly 2 meteors in the next 30 minutes?

Accepted Answer

Start by finding the average number of meteors for the window you actually care about. Thirty minutes is half an hour, so the mean count is \mu = \lambda t = 5 	imes 	frac12 = 2.5. On average you expect two and a half streaks in half an hour. The number seen, N, is Poisson with this mean, and the chance of exactly k is P(N=k) = e^{-\mu}\mu^k/k!. With k = 2: P(N = 2) = e^{-2.5}\,\frac{2.5^2}{2!} = e^{-2.5}\,\frac{6.25}{2} = 3.125\,e^{-2.5} \approx 3.125 	imes 0.0821 \approx 0.257. So roughly a 26\% chance of exactly two meteors. Takeaway. The rate can be any number, even a fraction like 2.5…

Two meteors in half an hour

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