Just the large raindrops

Question

Raindrops hit a sensor as a Poisson process at rate $50$ per minute. Independently, each drop is classified as "large" with probability $0.1$ .

What is the probability that exactly $3$ large drops hit the sensor in one minute?

Accepted Answer

Marking each drop independently as large or not is thinning. The large drops form their own Poisson process with rate \lambda{	ext{large}} = 50 	imes 0.1 = 5 	ext{ per minute}. Over one minute the mean number of large drops is \mu = 5 	imes 1 = 5. The chance of exactly 3 is P(N = 3) = e^{-5}\,\frac{5^3}{3!} = e^{-5}\,\frac{125}{6} \approx 20.83 	imes 0.006738 \approx 0.140. So about a 14\% chance. Whatever the small drops do, they do not affect the large-drop count. Takeaway. Splitting a Poisson stream by an independent label produces a Poisson sub-stream at rate \lambda p; each label li…

Just the large raindrops

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