Question

Cars arrive at a toll booth according to a Poisson process with mean 90 cars per hour. Suppose the attendant makes a phone call. How long, in seconds, can the attendant's phone call last if the probability is at least 0.1 that no cars arrive during the call

Answer 1

92.12 seconds

Explanation:

According to the poisson probability relation :

P(X =x) = (e^-λ * λ^x) / x!

For no calls to be reveived during the period, x = 0

P(X = 0) = (e^-λ * λ^0) / 0!

P(X = 0) = 0.1

0.1 = (e^-λ * λ^0) / 0!

0.1 = e^-λ

Take the In of both sides

In(0.1) = - λ

-2.303 = - λ

λ = 2.303

The length of call in second, l

l = λ / r ; r = arrival rate

r = 90 per hour ; this means ;

90 / 3600 = 0.025

l = 2.303 / 0.025

l = 92.12 seconds