PROBABILITY AND STATISTICS

1. PROBABILITY AND STATISTICS WEEK 6 Onur Doğan 2016-2017

2. Discrete Uniform Distribution A discrete uniform random variable X has an equal probability for each value inthe range ofX= [a, b], a < b. Thus, the probability massfunction of X is; P(x)= 1/(b-a+1) where x=a,a+1,…,b Onur Doğan 2016-2017

3. Example • Casting a die… Onur Doğan 2016-2017

4. Example Suppose that product codes of 2, 3, or 4 letters are equally likely. • Determine the probability mass function of the number of letters (X) in aproduct code. • Calculate the mean and variance ofX Onur Doğan 2016-2017

5. The Binomial Probability Distribution There are many experiments that conform either exactly or approximately to the followinglist of requirements: 1. The experiment consists of a sequence of n smaller experiments called trials,where n is fixed in advance of the experiment. 2. Each trial can result in one of the same two possible outcomes (dichotomoustrials), which we denote by success (S) and failure (F). 3. The trials are independent, so that the outcome on any particular trial does notinfluence the outcome on any other trial. 4. The probability of success is constant from trial to trial; we denote this probabilityby p. An experiment for which Conditions 1–4 are satisfied is called a binomialexperiment. Onur Doğan 2016-2017

6. Bernoulli trials A Bernoulli refers to a trial that has only two possible outcomes. (1) Flipping a coin: S = {head, tail) (2) Truth of an answer: S = {right, wrong) (3) Status of a machine: S = {working, broken) (4) Quality of a product: S = {good, defective) (5) Accomplishment of a task: S = {success, failure) A binomial experiment consists of a series of n independent Bernoulli trials with a constantprobability of success (p) in each trial. Onur Doğan 2016-2017

7. Example • A seller’s success ? Onur Doğan 2016-2017

8. The Mean and Variance of X Onur Doğan 2016-2017

9. Example • Suppose that a machine produce defective item with probability 0,1. a) Suppose that machineproduces5 items, find the probability of 1 item defective. b) Ifthe amount of daily production is 100, thenwhat's the expected defective item amount? c)What’s the variance of defective items of samples around the expected defective items. Onur Doğan 2016-2017

10. Example The probabilityof making a doctor's successful surgery is %80. Ifthat doctor make 3 surgery in one month, find the all probability for all possible results. Onur Doğan 2016-2017

11. Example • In a certain automobile dealership, 20% of all customers purchase an extended warranty with their new car. For 7 customers selected at random: 1) Find the probability that exactly 2 will purchase an extended warranty 2) Find the probability at most 6 will purchase an extended warranty Onur Doğan 2016-2017

12. Solutions: 1) n = 18, p = 0.75, q = 1 - 0.75 = 0.25 m = = = np ( 18 )(0. 75 ) 13 . 5 s = = = » npq ( 18 )(0. 75 )(0. 25 ) 3 . 375 1 . 8371 2) The probability function is: 18 æ ö - x 18 x = = P ( x ) (0. 75 ) (0. 25 ) for x 0, 1, 2, . . . , 18 ç ÷ è ø x Example • Example: Find the mean and standard deviation of the binomial distribution when n = 18 and p = 0.75. Define theprobabilityfunction. Onur Doğan 2016-2017

13. The Hypergeometric Distribution The assumptions leading to the hypergeometric distribution are as follows: • The population or set to be sampled consists of N individuals, objects, or elements(a finite population). 2. Each individual can be characterized as a success (S) or a failure (F), and thereare M successes in the population. 3. A sample of n individuals is selected without replacement in such a way thateach subset of size n is equally likely to be chosen. The random variable of interest is X the number of S’s in the sample. The probabilitydistribution of X depends on the parameters n, M, and N, so we wish toobtain P(X x) h(x; n, M, N). Onur Doğan 2016-2017

14. Example Suppose that a box contains five red balls and ten blueballs. If seven balls are selected at random without replacement,what is the probability that three redballs will be obtained? Onur Doğan 2016-2017

15. The Hypergeometric Distribution Onur Doğan 2016-2017

16. Example • Supposethat in productionlineforevery 20 products, 4 of thementerreprocessing. a) Ifweselected 2 products, findtheprobability of one of thementerreprocessing? b) Ifweselected 10 products, howmany of themshouldhaveexpectedenterreprocessing? Onur Doğan 2016-2017

17. Note: The hypergeometric distribution is related to the binomialdistribution. Whereas the binomial distribution is the approximate probabilitymodel for samplingwithout replacement from a finite dichotomous(S–F) population,the hypergeometricdistribution is the exact probability model for thenumberof S’s in the sample. Onur Doğan 2016-2017

18. TheNegativeBinomialDistribution The negative binomial rv and distribution are based on an experiment satisfying thefollowing conditions: 1. The experiment consists of a sequence of independent trials. 2. Each trial can result in either a success (S) or a failure (F). 3. The probability of success is constant from trial to trial, so P(S on trial i)=p fori=1, 2, 3 . . . . 4. The experiment continues (trials are performed) until a total of r successes havebeen observed, where r is a specified positive integer. Onur Doğan 2016-2017

19. TheNegativeBinomialDistribution Onur Doğan 2016-2017

20. Example A pediatrician wishes to recruit 5 couples, each of whom is expecting their first child,to participate in a new natural childbirth regimen. Let p=P(a randomly selected coupleagrees to participate). If p=0.2, what is the probability that 15 couples must beasked before 5 are found who agree to participate? Onur Doğan 2016-2017

21. TheNegativeBinomialDistribution Onur Doğan 2016-2017

22. The Geometric Distributions Onur Doğan 2016-2017

23. Example • In a production line 200 of 1000 items were found to be defective. a)What’s the probability of first defective item is the 4th item tested. b)How many itemsshould have been tested till first defective item found? c)What’s the probability of the first defective item is not the first tested one? Onur Doğan 2016-2017

24. The Multinomial Distributions Onur Doğan 2016-2017

25. Example Suppose that there are 3 different brand; A,B and C. And we have probabilities to be purchased; P(A)=0,40 P(B)=0,10 P(C)=0,50 Suppose that there are 10 customers, what’s the probability of 2 of them buy A, 5 of them buy B and 3 of them buy C. Onur Doğan 2016-2017

26. The Poisson Probability Distribution Onur Doğan 2016-2017

27. Example Suppose that,in İzmir the number ofpowerblackouthas the Poisson distributionwithmean 2, for oneyear. • Find the probability of there will be no power blackout in next year? • Find the probability of there will be 2 power blackout in next 6 months? • Find the probability of there will be 2 or more blackout in next year? Onur Doğan 2016-2017

28. Example The number of requests for assistance received by a towingservice is a Poisson process with rate=4 per hour. a. Compute the probability that exactly ten requests arereceived during a particular 2-hour period. b. If the operators of the towing service take a 30-min breakfor lunch, what is the probability that they do not miss any calls for assistance? c. How many calls would you expect during their break? Onur Doğan 2016-2017