Splitting an email stream into spam and real

Question

Emails hit your inbox as a Poisson process at rate $10$ per hour. Independently, each email is spam with probability $0.3$ and legitimate with probability $0.7$ .

What is the probability you receive exactly $2$ spam emails in a given hour? And are the spam count and legitimate count independent?

Accepted Answer

This is Poisson thinning. When you randomly color each point of a Poisson process (spam vs. legit) independently, the two colored sub-streams are themselves independent Poisson processes, with rates split according to the coloring probabilities: \lambda{	ext{spam}} = 10 	imes 0.3 = 3 	ext{ per hour}, \qquad \lambda{	ext{legit}} = 10 	imes 0.7 = 7 	ext{ per hour}. So the number of spam emails in an hour is Poisson with mean 3. The probability of exactly 2: P(N{	ext{spam}} = 2) = e^{-3}\frac{3^{2}}{2!} = e^{-3}\cdot\frac{9}{2} = 4.5\,e^{-3} \approx 4.5	imes 0.0498 \approx 0.224. About a…

Splitting an email stream into spam and real

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