1. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 1 MER301: Engineering Reliability LECTURE 5: Chapter 3: Probability Plotting,The Rules of Counting, Binomial Distribution, Poisson Distribution

2. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 2 Summary of Topics  Probability Plotting  Rules of Counting  Binomial Distribution  Poisson Distribution

3. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 3 Probability Plotting  Graphical method of testing a data set to see if it conforms to some specific distribution  For an ordered data set x(j) the Cumulative Distribution Function is plotted x(j)vs (j-0.5)/n where j=1,…,n  Probability plotting often used in failure analysis to predict future failures….

4. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 4 Normal Probability Plot

5. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 5 Normal Probability Plot

6. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 6 Normal Probability Plot

7. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 7 Rules of Counting  Fundamental Rule of Counting  Permutation Rules  Combinations

8. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 8 Rules of Counting  Fundamental Counting Rule The total number of ways a sequence of k events can occur with n i denoting the number of ways the i th event (i= 1 to k) can occur is  For buying a computer, there are choices of three hard drives n 1, two levels of RAM n 2, two video cards n 3, and three monitors n 4 so the total number of options is

9. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 9 Rules of Counting-con’t  Permutation Rules A permutation is an arrangement of n distinct objects in a specific order. The number of arrangements of n distinct objects in a specific order equals If the n objects are taken k at a time, then the number of arrangements in a specific order is A case where the order of selection is not important is called a Combination. The number of ways of selecting k objects from n objects without regard to order is  This is the permutation divided by k!, the number of ways the k objects can be arranged…

10. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 10 Rules of Counting-con’t  Combinations A case where the order of selection is not important is called a Combination. The number of ways of selecting k objects from n objects without regard to order is  This is the permutation divided by k!, the number of ways the k objects can be arranged… For the case where the objects are taken n at a time but some are identical, the number of possible arrangements in a specific order is

11. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 11 Rules of Counting-con’t  A Combination for n objects taken k at a time is equal to the Permutation of the n objects (n!/(n-k)! )divided by the number of ways the k objects can be arranged (k!)  For five objects taken three at a time

12. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 12 Binomial Distribution  The Binomial distribution describes the results of n independent identical success-failure trials Constant chance of a success or failure outcome(called probability p) Knowing the outcome of any one repetition does not change chance of any other repetition Must be able to count the number of successes and failures

13. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 13 Binomial Distribution

14. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 14 Binomial Distribution Combinations and Mean/Variance  From counting combinations, the number of combinations of n distinct objects selected x at a time is given by  Mean and Variance of the Binomial Distribution

15. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 15 Binomial Distributions for n=10 and p=0.1,0.5,0.9

16. L Berkley Davis Copyright 2009 Binomial Distributions for n=10 and p=0.1,0.5,0.9 MER301: Engineering Reliability Lecture 5 16 Excel formula for Binomial =binomdist(x,n,p,f(X=x)=false) =binomdist(x,n,p,cumulative=true)

17. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 17 Example 5.1  An electronics manufacturer claims that 10% or less of power supply units fail during warranty. To test this claim, an independent laboratory purchases 20 units and conducts accelerated life testing to measure failure during the warranty period. Let p denote the probability that a power supply unit fails during the testing period. The laboratory data resulting from testing will be compared to the claim that p  0.10. Let X denote the number among the 20 sampled that fail. What is the expected value and variance of X if the manufacturer’s claim is true? Find the probability that less than 5 of the 20 power supplies will fail if the manufacturer’s claim is true

18. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 18 Normal Approximation to the Binomial Distribution  For values of np>5 and n(1-p)>5, probabilities from the binomial distribution can be approximated as follows:

19. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 19 Poisson Distribution  The Poisson Distribution gives the predicted probability of a specific number of events occurring in an interval of known size, when the mean number of events in such intervals is known Number of goals scored in a game Errors in transmission of data

20. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 20 Poisson Distribution  Assumptions for Poisson Distribution Random discrete events that occur in an interval that can be divided into subintervals Probability of a single occurrence of the event is directly proportional to the size of a subinterval If the sampling subinterval is sufficiently small, the probability of two or more occurrences of the event is negligible Occurrences of the event in nonoverlapping subintervals are independent

21. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 21 Poisson Distribution

22. L Berkley Davis Copyright 2009 Poisson Examples:lambda =1,4.5,9 MER301: Engineering Reliability Lecture 5 22 Excel Formula for Poisson =poisson(x,lambda,f(X=x)=false) =poisson(x,lambda,cumulative=true)

23. L Berkley Davis Copyright 2009 Poisson Examples:lambda =1,4.5,9 MER301: Engineering Reliability Lecture 5 23

24. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 24 Poisson Process  Observing discrete events in a continuous “interval” of time, length or space The number of white blood cells in a drop of blood The number of times excessive pollutant levels are emitted from a gas turbine power plant during a three-month period

25. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 25 Example 5.2  Yeast is added when mash is prepared for fermentation in the beer making process. The yeast is cultured in vats and the exact amount of yeast added to the mash is of critical importance. The yeast cell count in culture fluid averages 6000 yeast cells per cubic millimeter of fluid in the culture vats. It is however necessary to know the concentration in a specific vat to know how much fluid to add to the mash. The distribution of yeast cells is known to follow a Poisson Distribution. To establish the yeast cell concentration, a 0.001 cubic millimeter drop is taken and the number of yeast cells X is counted What is the probability of getting two or less yeast cells in a single sample? What is the probability of getting more than two yeast cells in a single sample? How many yeast cells are expected if the sample is from a typical culture vat?

26. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 26 Normal Approximation of the Poisson Distribution  Poisson developed for case of n approaching infinity For mean >5 then:

27. L Berkley Davis Copyright 2009 MER301: Engineering Reliability Lecture 5 27 Summary of Topics  Probability Plotting  Rules of Counting  Binomial Distribution  Poisson Distribution