Which visitors actually subscribe

Question

Visitors reach a website as a Poisson process at rate $200$ per hour. Independently, each visitor subscribes with probability $0.02$ .

What is the probability of exactly $2$ new subscribers in the next $30$ minutes?

Accepted Answer

Each visitor independently subscribes or not, which thins the visitor stream down to a subscriber stream. The subscribers form a Poisson process with rate \lambda{	ext{sub}} = 200 	imes 0.02 = 4 	ext{ per hour}. Thirty minutes is half an hour, so the mean number of subscribers is \mu = \lambda{	ext{sub}}\, t = 4 	imes 	frac12 = 2. The chance of exactly 2 subscribers is P(N = 2) = e^{-2}\,\frac{2^2}{2!} = 2\,e^{-2} \approx 0.271. About a 27\% chance. The overwhelming majority of visitors who did not subscribe are simply irrelevant to this count. Takeaway. A small conversion probability j…

Which visitors actually subscribe

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