Question

Introduction to Poisson and Exponential distribution, Memoryless Property

You are working to statistically model the location of defects on the surface of a 3D printed material. After analyzing for one month you found out that the number of defects in the material follows a Poisson process, with an average of one defect is found every 10µm² area. One of your lab mates is willing to validate the information. He selects 60 µm² are on the surface to study the location of defects. Using this information provided, answer questions 1 to 4.

1. What is the expected number of defects that your lab mate will find on the area he is studying?

2. What is the probability that your lab mate finds at least 3 defects on the surface area he is observing(round to 3 digits)?

3. Your lab mate just found one defect. What is the probability that he will find another defect in less than 4 µm² area nearby (round to 4 digits)?

4. Your lab mate finds that he has searched about 5 µm² area since the last defect. What is the probability that he has to search another 8 µm² are to find the next defect(round to 3 digits)?

Answer 1

1)

X = number of defects found in 60 µm²

X follow poisson distribution with = 1/10 * 60 = 6

E(X) = = 6

2)

P(X >= 3)

=1 - P(X <=2)

= =1-poisson(2,6,1)

=0.9380

3)

probability of find another defect in less than 4 µm² area = 1 - probability of not finding in less than 4 µm²

= 1 - e^(-4/10)

= 0.3297

4)

Inter-arrival follow exponential distribution

P(X > 13 | X > 5)

= P(X > 8)            {exponential distribution have memoryless property}

= e^(-8/10)

= 0.4493