Stochastic Process → Poisson Process → Definition → Examples → Arrival and Inter-Arrival Times

**Probability of the first Arrival**

Let *N*(*t*) be a Poisson process with rate . Let *X*_{1} be the time of the first arrival. Then probability of the first arrival is given by

**Example** An office receive telephone calls according to a Poisson process at mean rate of five per hour. The office timing is from 9:00 to 5:00.

**(a)** What is the probability that the first call receive after 9:30?

**(b)** The lunch break is from 12:00 to 12:30. What is the probability that after the lunch break, the first call receive after 1:30?

**(c)** What is the probability that after 12:30, the first message arrive by 2:00?

**Solution**

**(a)** Here λ = 5 and *t* = 0.5 hour. So, using the above formula

**(b)** Here λ = 5 and *t* = 1 hour (starting from 12:30). Hence,

**(c)** Here λ = 5 and *t* = 1.5 hour (after 12:30). Hence,

**Remember:** “Arrive by” means less than or equal to (≤).

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