Question

Emails arrive in an inbox according to a Poisson process with rate emails per hour. Let be the time at which the third email arrives, measured in hours after a certain fixed starting time. Find without using calculus

Answer 1

Answer:

Find P(T>0.1)P(T>0.1) without using calculus.

Note that P(T>0.1)P(T>0.1) = 1−P(T<0.1)1−P(T<0.1).

Let X1,X2,X3X1,X2,X3 be iid Expo(20)Expo(20), then T=X1+X2+X3T=X1+X2+X3

Clearly, TT follows a gamma distribution. with λ=20λ=20 and k=3k=3. But in order to evaluate the solution using this way, calculus is needed.

Now the question: How can this question be solved w/o calculus? (e.g., by some kind of story/analogy)