Poisson hits on a website

Question

A website receives hits according to a Poisson process at $120$ per hour, that is, $2$ per minute.

(a) What is the probability of at least one hit in the next $30$ seconds? (b) What is the expected wait until the next hit?

Accepted Answer

(a) In 30 seconds = 	frac12 minute the count is Poisson with mean \mu = 2 	imes 	frac12 = 1. "At least one" is easiest via the complement: P(N \ge 1) = 1 - P(N = 0) = 1 - e^{-1} \approx 1 - 0.368 = 0.632. (b) The gap between consecutive hits is exponential with rate \lambda = 2 per minute, whose mean is \frac{1}{\lambda} = 	frac12 minute = 30 seconds. So you wait about 30 seconds on average. Note the two answers are two views of the same \mu = 1: a mean of one hit per half-minute means both "roughly a 63\% chance of seeing one" and "about half a minute between hits." Takeaway. For "at lea…

Poisson hits on a website

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