Expected time to the twenty-fifth email

Question

Emails arrive as a Poisson process at rate \lambda = 10 per hour. On average, how long (in minutes) until the 25th email of the day arrives?

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The time until the k-th Poisson arrival is a sum of k independent interarrival waits, each with mean 1/\lambda. So the expected time to the k-th arrival is just k of those mean waits added together: E[	ext{time to } k	ext{-th arrival}] = \frac{k}{\lambda}. With k = 25 and \lambda = 10 per hour, E = \frac{25}{10} = 2.5 	ext{ hours} = 150 	ext{ minutes}. So on average the 25th email lands about 150 minutes in. The intuition: each email takes 1/\lambda = 6 minutes on average, and 25 of them stack up to 150 minutes. Takeaway. The expected time to the k-th event is simply k/\lambda, that is k …

Expected time to the twenty-fifth email

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