1. 1 1 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Chapter 5: Probability Distributions: Discrete Probability Distributions

2. 2 2 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Learning Objectives n Identifying Types of Discrete Probability Distribution and their Respective Functional Representations n Calculating the Mean and Variance of a Discrete Random Variable with each of the different discrete probability distribution identified.

3. 3 3 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Random Variable

4. 4 4 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Random Variables n Any variable that is used to represent the outcomes any experiment of interest to us is called Random Variable. n A random variable can assume (take) any value (Positive, negative, zero; finite, infinite; continuous and discrete values). n Depending upon the values it takes, we can identify two types of random variables: 1. Discrete Variables 2. Continuous Variables.

5. 5 5 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Note that both types of v ariables can assume either a finite number of values or an infinite sequence of values. Note that both types of v ariables can assume either a finite number of values or an infinite sequence of values.

6. 6 6 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Let x = number of TVs sold at a given store in one day. Let x = number of TVs sold at a given store in one day. The number of TV units that can be sold in a given day is finite. It is also discrete: (0, 1, 2, 3, 4). X can be considered as Discrete Random Variable Let x = number of TVs sold at a given store in one day. Let x = number of TVs sold at a given store in one day. The number of TV units that can be sold in a given day is finite. It is also discrete: (0, 1, 2, 3, 4). X can be considered as Discrete Random Variable Example: JSL Appliances n Discrete random variable with a finite number of values

7. 7 7 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Let Y = number of customers arriving in a store in one day. Let Y = number of customers arriving in a store in one day. Y can take on the values 0, 1, 2,... Y can take on the values 0, 1, 2,... Let Y = number of customers arriving in a store in one day. Let Y = number of customers arriving in a store in one day. Y can take on the values 0, 1, 2,... Y can take on the values 0, 1, 2,... Example: JSL Appliances n Discrete random variable with an infinite sequence of values We can count the customers arriving. However, there is no finite upper limit on the number that might arrive.

8. 8 8 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Random Variables Question Random Variable x Type Familysize X = Number of dependents reported on tax return reported on tax returnDiscrete Distance from home to a store Y = Distance in miles from home to the store site home to the store site Continuous Owning dog or cat Z = 1 if own no pet; = 2 if own dog(s) only; = 2 if own dog(s) only; = 3 if own cat(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) = 4 if own dog(s) and cat(s) Discrete

9. 9 9 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Probability Distributions

10. 10 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The probability distribution for a random variable is a distribution The probability distribution for a random variable is a distribution that describes the values that the random variable of interest takes. The probability distribution for a random variable is a distribution The probability distribution for a random variable is a distribution that describes the values that the random variable of interest takes. Probability Distributions It is defined by a probability function, and is denoted by f ( x ) It is defined by a probability function, and is denoted by f ( x ) ---the probability of the values of the random variable. It is defined by a probability function, and is denoted by f ( x ) It is defined by a probability function, and is denoted by f ( x ) ---the probability of the values of the random variable.

11. 11 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) For any probability function the following conditions must be satisfied: satisfied: Probability Distributions 1.f ( x ) > 0  f ( x ) = 1

12. 12 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Discrete Probability Distributions

13. 13 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) A Discrete Probability Distribution is a tabular, graphic or Functional representation of a Random Variable with discrete outcomes that follows the principle of probability distribution A Discrete Probability Distribution is a tabular, graphic or Functional representation of a Random Variable with discrete outcomes that follows the principle of probability distribution Discrete Probability Distributions f ( x ) > 0  f ( x ) = 1

14. 14 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Using past data on sales, a tabular representation of the probability distribution for TV sales was developed.Using past data on sales, a tabular representation of the probability distribution for TV sales was developed. Number Number Units Sold of Days Units Sold of Days 0 80 1 50 1 50 2 40 2 40 3 10 3 10 4 20 4 20 200 200 x f ( x ) x f ( x ) 0.40 0.40 1.25 1.25 2.20 2.20 3.05 3.05 4.10 4.10 1.00 1.00 80/200 Discrete Probability Distributions--Example

15. 15 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse).10.20.30. 40.50 0 1 2 3 4 Values of Random Variable x (TV sales) ProbabilityProbability Discrete Probability Distributions n Graphical Representation of Probability Distribution

16. 16 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 1. Uniform Distribution 2. Binomial Distribution 3. Poisson Distribution There are four basic types of discrete probability distributions: 4. Hyper-Geometric Distribution

17. 17 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 5.1) Discrete Uniform Probability Distribution The probability distribution of a discrete probability The probability distribution of a discrete probability distribution is given by the following formula. distribution is given by the following formula. The probability distribution of a discrete probability The probability distribution of a discrete probability distribution is given by the following formula. distribution is given by the following formula. f ( x ) = 1/ n where: n = the number of values the random variable may assume variable may assume the values of the random variable random variable are equally likely are equally likely

18. 18 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) where: where: f ( x ) = the probability of x successes in n trials f ( x ) = the probability of x successes in n trials n = the number of trials n = the number of trials p = the probability of success on any one trial p = the probability of success on any one trial 5.2) Binomial Distribution n The probability distribution of a Binomial Distribution is given by the following function

19. 19 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n The probability of a Poisson Distribution is given by the following function 5.3) Poisson Distribution where: where: f(x) = probability of x occurrences in an interval  = mean number of occurrences in an interval  = mean number of occurrences in an interval e = 2.71828 e = 2.71828

20. 20 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n The probability of a Hyper-geometric Distribution is given by the following Function 5.4) Hypergeometric Distribution for 0 < x < r where: f ( x ) = probability of x successes in n trials n = number of trials n = number of trials N = number of elements in the population N = number of elements in the population r = number of elements in the population r = number of elements in the population labeled success labeled success

21. 21 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 5.1) Discrete Uniform Probability Distribution The probability distribution of a discrete probability The probability distribution of a discrete probability distribution is given by the following formula. distribution is given by the following formula. The probability distribution of a discrete probability The probability distribution of a discrete probability distribution is given by the following formula. distribution is given by the following formula. f ( x ) = 1/ n where: n = the number of values the random variable may assume variable may assume the values of the random variable random variable are equally likely are equally likely

22. 22 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Expected Value (Mean) and Variance of Discrete Probability Distributions

23. 23 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Expected Value (Mean) for … …..Discrete Uniform Distribution Expected Value (Mean) for … …..Discrete Uniform Distribution The expected value, or mean, of a random variable The expected value, or mean, of a random variable is a measure of its central location. is a measure of its central location. The expected value, or mean, of a random variable The expected value, or mean, of a random variable is a measure of its central location. is a measure of its central location. E ( x ) =  =  xf ( x )

24. 24 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Variance and Standard Deviation of Discrete Uniform Probability Distribution The variance summarizes the variability in the values of a random The variance summarizes the variability in the values of a random variable. variable. The variance summarizes the variability in the values of a random The variance summarizes the variability in the values of a random variable. variable. The standard deviation, , is defined as the square root of the The standard deviation, , is defined as the square root of thevariance. variance. Var( x ) =  2 =  ( x -  ) 2 f ( x )

25. 25 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The likelihood of selling TV sets in any given day is considered equally likely (Uniform). The following table summarizes sales data on the past 200 days.The likelihood of selling TV sets in any given day is considered equally likely (Uniform). The following table summarizes sales data on the past 200 days. Number Number Units Sold of Days Units Sold of Days 0 80 1 50 1 50 2 40 2 40 3 10 3 10 4 20 4 20 200 200 x f ( x ) x f ( x ) 0.40 0.40 1.25 1.25 2.20 2.20 3.05 3.05 4.10 4.10 Example-TV Sales in a Given Store n Given this data what is the Average Number of TVs sold in a day? x F ( x ) x F ( x ) 0.40 0.40 1.65 1.65 2.85 2.85 3.90 3.90 4 1.00 4 1.00

26. 26 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Given the data, we can find the Expected Value (Mean Number) of TVs sold in a day as follows expected number of TVs sold in a day x f ( x ) xf ( x ) x f ( x ) xf ( x ) 0.40.00 0.40.00 1.25.25 1.25.25 2.20.40 2.20.40 3.05.15 3.05.15 4.10.40 4.10.40 E ( x ) = 1.20 E ( x ) = 1.20 Expected Value (MEAN) and Variance E ( x ) =  =  xf ( x )

27. 27 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 01234 -1.2-0.2 0.8 0.8 1.8 1.8 2.8 2.81.440.040.643.247.84.40.25.20.05.10.576.010.128.162.784 x -  ( x -  ) 2 f(x)f(x)f(x)f(x) ( x -  ) 2 f ( x ) x TVs squared Standard deviation of daily sales = 1.2884 TVs Find the Variance and Standard Deviation of the Number of TVs Sold in a given day. Var( x ) =  2 =  ( x -  ) 2 f ( x )= 1.66 Var( x ) =  2 =  ( x -  ) 2 f ( x )

28. 28 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 5.2) The Binomial Distribution

29. 29 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 5.2) Binomial Distribution n A discrete probability distribution with binomial distribution has the following propertie 3. The probability of a success, denoted by p, does not change from trial to trial. not change from trial to trial. 3. The probability of a success, denoted by p, does not change from trial to trial. not change from trial to trial. 4. The trials are independent. 2. Only two outcomes, success and failure, are possible on each trial. on each trial. 2. Only two outcomes, success and failure, are possible on each trial. on each trial. 1. The experiment consists of a sequence of n identical trials. identical trials. 1. The experiment consists of a sequence of n identical trials. identical trials.

30. 30 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 5.2) The Binomial Distribution Typical Examples of a Binomial Experiment: Lottery: Win or Lose Lottery: Win or Lose Election: A Candidate Wins or Loses Election: A Candidate Wins or Loses Gender of an Employee: is Male or Female Gender of an Employee: is Male or Female Flipping a coin: Heads or Tails Flipping a coin: Heads or Tails

31. 31 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 5.2) The Binomial Distribution Our interest is in the number of successes occurring in the n trials. Our interest is in the number of successes occurring in the n trials. let X denote the number of successes occurring in the n trials.

32. 32 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) where: where: n = the number of trials p = the probability of success on any one trial p = the probability of success on any one trial f(x) = the probability of x successes in n trials f(x) = the probability of x successes in n trials 5.2) The Binomial Distribution n Binomial Probability Function

33. 33 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 5.2) The Binomial Distribution n Binomial Probability Function Probability of a particular sequence of outcomes sequence of outcomes with x successes with x successes Probability of a particular sequence of outcomes sequence of outcomes with x successes with x successes Number of experimental outcomes providing exactly outcomes providing exactly x successes in n trials Number of experimental outcomes providing exactly outcomes providing exactly x successes in n trials

34. 34 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Binomial Distribution n Example: Evans Electronics A local Electronics company is concerned about its low retention of employees. In recent years, management has seen an annual turnover of 10% in its hourly employees. Thus, for any hourly employee chosen at random, the company estimates that there is 0.1 probability that the person will leave the company in a year time.

35. 35 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Binomial Distribution n Given the above information, if we randomly select 3 hourly employees, what is the probability that 1 of them will leave the company in one year ? Let : p =.10, n = 3, x = 1 Solution:

36. 36 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Tree Diagram Binomial Distribution 1 st Worker 2 nd Worker 3 rd Worker x x Prob. Leaves (.1) Leaves (.1) Stays (.9) Stays (.9) 3 3 2 2 0 0 2 2 2 2 Leaves (.1) S (.9) Stays (.9) S (.9) L (.1).0010.0090.7290.0090 1 1 1 1.0810 11

37. 37 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Using Tables of Binomial Probabilities Binomial Distribution Page….592-600

38. 38 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Mean and Variance of A Binomial Distribution E ( x ) =  = np Var( x ) =  2 = np (1  p ) n Expected Value n Variance n Standard Deviation

39. 39 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) E ( x ) =  = 3(.1) =.3 employees out of 3 Var( x ) =  2 = 3(.1)(.9) =.27 n Expected Value n Variance n Standard Deviation Given that p=0.1, for the 3 randomly selected hourly employees, what is the expected number and variance of workers who might leave the company this year?

40. 40 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 5.3) Poisson Distribution

41. 41 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Poisson distribution refers to the probability distribution of a trial that involves cases of rare events that occur over a fixed time interval or within a specified region 6.4. Poisson Distribution

42. 42 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Examples…. The number of errors a typist makes per page The number of errors a typist makes per page The number of cars entering a service station per hour The number of cars entering a service station per hour The number of telephone calls received by a switchboard per hour. The number of telephone calls received by a switchboard per hour. The number of bank failures during a given economic recession. The number of bank failures during a given economic recession. The number of housing foreclosures in a given city during a given year. The number of housing foreclosures in a given city during a given year. The number of car accidents in one day on I-35 stretch from the Twin Cities to Duluth The number of car accidents in one day on I-35 stretch from the Twin Cities to Duluth 6.4. Poisson Distribution

43. 43 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) We use a Poisson distribution to estimate the number of We use a Poisson distribution to estimate the number of occurrences of these discrete events that often occur over a specified interval of time or space a specified interval of time or space We use a Poisson distribution to estimate the number of We use a Poisson distribution to estimate the number of occurrences of these discrete events that often occur over a specified interval of time or space a specified interval of time or space That is, a Poisson distributed random variable is discrete; Often times it assumes an infinite sequence of values (x = 0, 1, 2,... ). That is, a Poisson distributed random variable is discrete; Often times it assumes an infinite sequence of values (x = 0, 1, 2,... ). Poisson Distribution

44. 44 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n The number of successes (events) that occur in a certain time interval is independent of the number of successes that occur in another time interval. n The probability of a success in a certain time interval is the same for all time intervals of the same size; proportional to the length of the interval. the same for all time intervals of the same size; proportional to the length of the interval. –The probability that two or more successes will occur in an interval approaches zero as the interval becomes smaller. Properties of a Poisson Experiment

45. 45 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Poisson Probability Function Poisson Distribution where: where: f(x) = probability of x occurrences in an interval  = mean number of occurrences in an interval  = mean number of occurrences in an interval e = 2.71828 e = 2.71828

46. 46 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) On average 6 patients arrive per hour at Mercy Hospital per hour at Mercy Hospital emergency room on weekend emergency room on weekend evenings. evenings. What is the probability of 4 arrivals in 30 minutes on a 4 arrivals in 30 minutes on a weekend evening? Poisson Distribution--Example MERCY

47. 47 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Poisson Distribution-Example n Using the Poisson Probability Function  = 6/hour = 3/half-hour, P( x = 4)?

48. 48 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Poisson Distribution n Using Poisson Probability Tables Pages: 602-607

49. 49 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Poisson Distribution of Arrivals Poisson Distribution Poisson Probabilities 0.00 0.05 0.10 0.15 0.20 0.25 0 12345678910 Number of Arrivals in 30 Minutes Probability actually, the sequence continues: 11, 12, …

50. 50 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Mean and Variance of Poisson Distribution Another special property of the Poisson distribution is that the mean and variance are equal.  Another special property of the Poisson distribution is that the mean and variance are equal.   =  2

51. 51 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Poisson Distribution n Variance for Number of Arrivals During 30-Minute Periods  =  2 = 3

52. 52 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 5.4) Hyper-Geometric Distribution

53. 53 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution The hyper-geometric distribution is closely related The hyper-geometric distribution is closely related to a binomial distribution. to a binomial distribution. The hyper-geometric distribution is closely related The hyper-geometric distribution is closely related to a binomial distribution. to a binomial distribution. However, for the hyper-geometric distribution: However, for the hyper-geometric distribution: the trials are not independent, and the trials are not independent, and the probability of success changes from trial the probability of success changes from trial to trial. to trial. the probability of success changes from trial the probability of success changes from trial to trial. to trial.

54. 54 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Hyper-geometric Probability Function Hyper-geometric Distribution for 0 < x < r where: f ( x ) = probability of x successes in n trials n = number of trials n = number of trials N = number of elements in the population N = number of elements in the population r = number of elements labeled as success r = number of elements labeled as success

55. 55 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Hyper-geometric Probability Function Hyper-geometric Distribution for 0 < x < r number of ways n – x failures can be selected from a total of N – r failures in the population number of ways x successes can be selected from a total of r successes in the population number of ways a sample of size n can be selected from a population of size N

56. 56 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution n Mean n Variance

57. 57 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution: Inspection Electric fuses produced by a given company are packed in boxes. Each box carries 12 units of electric fuses. The role of an inspector in the company is to make sure that all fuses in each box are in good condition. Consider the following scenario: A worker inadvertently places 5 defective items in a box. As part of her job, the inspector randomly selects 3 fuses from the box that contains the defective fuses.

58. 58 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution: Inspection 1. What is the probability that none of the three randomly selected fuses are defective? 2. What is the probability that the inspector finds only one of the three randomly selected fuses to be defective? 3. What is the probability that the inspector finds at least one of the three fuses defective?

59. 59 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution: Inspection 1. What is the probability that none of the three randomly selected fuses are defective? Solution: N=12; r=5; n=3; x=0; P(x=0)?

60. 60 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution: Inspection 2.What is the probability that the inspector finds only one of the three randomly selected fuses to be defective? Solution: N=12; r=5; n=3; x=1; P(x=1)?

61. 61 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution: Inspection 2. What is the probability that the inspector finds at least one of the three fuses defective? Solution: N=12; r=5; n=3; x 1

62. 62 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution n Example: Neveready Bob has removed two dead batteries from a flashlight and inadvertently mingled them with the two good batteries that he intended to use as replacements. The four batteries look identical. Then Bob randomly selects two of the four batteries. What is the probability that he selects the two good batteries? ZAP ZAP ZAP ZAP

63. 63 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution n Using the Hyper-geometric Function N = 4 = number of batteries in total r = 2 = number of good batteries in total r = 2 = number of good batteries in total x = 2 =number of good batteries to be selected x = 2 =number of good batteries to be selected n = 2 = number of batteries selected n = 2 = number of batteries selected

64. 64 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution n Mean n Variance

65. 65 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution n Mean n Variance

66. 66 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution When the population size is large, a hyper-geometric When the population size is large, a hyper-geometric distribution can be approximated by a binomial distribution can be approximated by a binomial distribution with n trials and a probability of success distribution with n trials and a probability of success p = ( r / N ). p = ( r / N ). When the population size is large, a hyper-geometric When the population size is large, a hyper-geometric distribution can be approximated by a binomial distribution can be approximated by a binomial distribution with n trials and a probability of success distribution with n trials and a probability of success p = ( r / N ). p = ( r / N ).

67. 67 Slide University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Hyper-geometric Distribution Consider a hyper-geometric distribution with n trials and let p = ( r / n ) denote the probability of a success and let p = ( r / n ) denote the probability of a success on the first trial. on the first trial. If the population size is large, the term ( N – n )/( N – 1) approaches 1. approaches 1. he expected value and variance can be written The expected value and variance can be written E ( x ) = np and Var ( x ) = np (1 – p ). E ( x ) = np and Var ( x ) = np (1 – p ). he expected Note that these are the expressions for the expected value and variance of a binomial distribution. value and variance of a binomial distribution.