Question

For a particular intersection, it has been bserved that the rate of people going thorugh the intersection while talking on their cell phone follows a Poisson process with an average of 6 people talking on their cell ohone per minute. Let X = number of people talking on their cell phone while driving through this intersection every 5 minutes. What is the probability that the 15th person talking on their cell phone will be observed going through this intersection in at most 3 minutes?

Answer 1

GIVEN:

The rate of people going through the intersection while talking on their cell phone follows a Poisson process with an average of 6 people talking on their cell phone per minute.

​ people per minute.

Let X be the number of people talking on their cell phone while driving through this intersection every 5 minutes with average number of people ​ people per 5 minutes.

TO FIND:

The probability that the 15th person talking on their cell phone will be observed going through this intersection in at most 3 minutes.

SOLUTION:

During 3 minutes period, the average number of people talking on their cell phone​ is ​​ people per 3 minutes (by converting average number of people per 5 minutes to average number of people per 3 minutes). Now let X be the the number of people talking on their cell phone while driving through this intersection every 3 minutes with average number of people talking on their cell phones is 18 people per 3 minutes.

The PDF of poisson distribution is given by,

Thus the probability that the 15th person talking on their cell phone observed going through this intersection in at most 3 minutes is given by,

Thus the probability that the 15th person talking on their cell phone will be observed going through this intersection in at most 3 minutes is 0.079.