Homework Assignment 3 Summer 2018 Question 1: Poisson processes Part 1 You have

Question

Homework Assignment 3 Summer 2018 Question 1: Poisson processes Part 1 You have been tasked by the city of Greensboro to fix a speeding camera located on Elm and Murrow. To fix the camera, you need to turn it off for 90 minutes. The camera catches people breaking the speed limit with a rate of A-1 per hour. Answer the following questions (a) What is the probability no speeding cars pass by while you are fixing the camera? (b) You decide to do an experiment: wait for the next speeding car to pass, and when they do, you will then turn off the camera to fix it. What is the probability no speeding car will pass by now? (c) There are three types of failures: one that is easily fixed in the 90 minutes of down time, one that will require an additional 30 minutes, and one that will requirean aditional 90 minutes What is the probability no speeding cars will pass by under these conditions? Answer:

Explanation / Answer

Sol:

(a)

X ~ Poisson(1 per hour) represents the number of cars violating the speed limit

L = average poisson arrival rate per 90 minutes = 1 x 90/60 = 1.5 per 90 minutes

P(X=0) = e-1.5 x 1.50 / 0! = 0.223

(b)

The forgetfulness property of Poisson process ensures that probabilities of future occurrences are uninfluenced by past events.

So, the probability will be uninfluenced by the fact when the fixing of the camera started. It will still be 0.223.

(c)

For 90 minutes, the probability is already found

For 90+30=120 minutes, P(X=0) = e-2.0 x 2.00 / 0! = 0.135

For 90+90=180 minutes, P(X=0) = e-3.0 x 3.00 / 0! = 0.050

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