1. 1 1 Slide © 2006 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University

2. 2 2 Slide © 2006 Thomson/South-Western Chapter 5 Discrete Probability Distributions.10.20.30.40 0 1 2 3 4 n Random Variables n Discrete Probability Distributions n Expected Value and Variance n Binomial Distribution n Poisson Distribution n Hypergeometric Distribution

3. 3 3 Slide © 2006 Thomson/South-Western A random variable is a numerical description of the A random variable is a numerical description of the outcome of an experiment. outcome of an experiment. A random variable is a numerical description of the A random variable is a numerical description of the outcome of an experiment. outcome of an experiment. Random Variables A discrete random variable may assume either a A discrete random variable may assume either a finite number of values or an infinite sequence of finite number of values or an infinite sequence of values. values. A discrete random variable may assume either a A discrete random variable may assume either a finite number of values or an infinite sequence of finite number of values or an infinite sequence of values. values. A continuous random variable may assume any A continuous random variable may assume any numerical value in an interval or collection of numerical value in an interval or collection of intervals. intervals. A continuous random variable may assume any A continuous random variable may assume any numerical value in an interval or collection of numerical value in an interval or collection of intervals. intervals.

4. 4 4 Slide © 2006 Thomson/South-Western Let x = number of TVs sold at the store in one day, Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) where x can take on 5 values (0, 1, 2, 3, 4) Let x = number of TVs sold at the store in one day, Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) where x can take on 5 values (0, 1, 2, 3, 4) Example: JSL Appliances n Discrete random variable with a finite number of values

5. 5 5 Slide © 2006 Thomson/South-Western Let x = number of customers arriving in one day, Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... where x can take on the values 0, 1, 2,... Let x = number of customers arriving in one day, Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... where x 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, but there is no We can count the customers arriving, but there is no finite upper limit on the number that might arrive.

6. 6 6 Slide © 2006 Thomson/South-Western Random Variables Question Random Variable x Type Familysize x = Number of dependents reported on tax return reported on tax returnDiscrete Distance from home to store x = Distance in miles from home to the store site home to the store site Continuous Own dog or cat x = 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

7. 7 7 Slide © 2006 Thomson/South-Western The probability distribution for a random variable The probability distribution for a random variable describes how probabilities are distributed over describes how probabilities are distributed over the values of the random variable. the values of the random variable. The probability distribution for a random variable The probability distribution for a random variable describes how probabilities are distributed over describes how probabilities are distributed over the values of the random variable. the values of the random variable. We can describe a discrete probability distribution We can describe a discrete probability distribution with a table, graph, or equation. with a table, graph, or equation. We can describe a discrete probability distribution We can describe a discrete probability distribution with a table, graph, or equation. with a table, graph, or equation. Discrete Probability Distributions

8. 8 8 Slide © 2006 Thomson/South-Western The probability distribution is defined by a The probability distribution is defined by a probability function, denoted by f ( x ), which provides probability function, denoted by f ( x ), which provides the probability for each value of the random variable. the probability for each value of the random variable. The probability distribution is defined by a The probability distribution is defined by a probability function, denoted by f ( x ), which provides probability function, denoted by f ( x ), which provides the probability for each value of the random variable. the probability for each value of the random variable. The required conditions for a discrete probability The required conditions for a discrete probability function are: function are: The required conditions for a discrete probability The required conditions for a discrete probability function are: function are: Discrete Probability Distributions f ( x ) > 0  f ( x ) = 1

9. 9 9 Slide © 2006 Thomson/South-Western n a tabular representation of the probability distribution for TV sales was developed. distribution for TV sales was developed. n Using past data on TV sales, … 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

10. 10 Slide © 2006 Thomson/South-Western.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

11. 11 Slide © 2006 Thomson/South-Western Discrete Uniform Probability Distribution The discrete uniform probability distribution is the The discrete uniform probability distribution is the simplest example of a discrete probability simplest example of a discrete probability distribution given by a formula. distribution given by a formula. The discrete uniform probability distribution is the The discrete uniform probability distribution is the simplest example of a discrete probability simplest example of a discrete probability distribution given by a formula. distribution given by a formula. The discrete uniform probability function is The discrete uniform probability function is 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

12. 12 Slide © 2006 Thomson/South-Western Expected Value and Variance 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. The variance summarizes the variability in the The variance summarizes the variability in the values of a random variable. values of a random variable. The variance summarizes the variability in the The variance summarizes the variability in the values of a random variable. values of a random variable. The standard deviation, , is defined as the positive The standard deviation, , is defined as the positive square root of the variance. square root of the variance. The standard deviation, , is defined as the positive The standard deviation, , is defined as the positive square root of the variance. square root of the variance. Var( x ) =  2 =  ( x -  ) 2 f ( x ) E ( x ) =  =  xf ( x )

13. 13 Slide © 2006 Thomson/South-Western n Expected Value 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 and Variance

14. 14 Slide © 2006 Thomson/South-Western n Variance and Standard Deviation 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 ) Variance of daily sales =  2 = 1.660 x TVssquared Standard deviation of daily sales = 1.2884 TVs Expected Value and Variance

15. 15 Slide © 2006 Thomson/South-Western Binomial Distribution n Four Properties of a Binomial Experiment 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. Two outcomes, success and failure, are possible on each trial. on each trial. 2. 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. stationarityassumption

16. 16 Slide © 2006 Thomson/South-Western Binomial Distribution Our interest is in the number of successes Our interest is in the number of successes occurring in the n trials. occurring in the n trials. Our interest is in the number of successes Our interest is in the number of successes occurring in the n trials. occurring in the n trials. We let x denote the number of successes We let x denote the number of successes occurring in the n trials. occurring in the n trials. We let x denote the number of successes We let x denote the number of successes occurring in the n trials. occurring in the n trials.

17. 17 Slide © 2006 Thomson/South-Western 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 Binomial Distribution n Binomial Probability Function

18. 18 Slide © 2006 Thomson/South-Western Binomial Distribution n Binomial Probability Function Probability of a particular sequence of trial outcomes sequence of trial outcomes with x successes in n trials with x successes in n trials Probability of a particular sequence of trial outcomes sequence of trial outcomes with x successes in n trials with x successes in n trials 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

19. 19 Slide © 2006 Thomson/South-Western Binomial Distribution n Example: Evans Electronics Evans is concerned about a low retention rate for employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year. Evans is concerned about a low retention rate for employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year.

20. 20 Slide © 2006 Thomson/South-Western Binomial Distribution n Using the Binomial Probability Function Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year? Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year? Let : p =.10, n = 3, x = 1

21. 21 Slide © 2006 Thomson/South-Western 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

22. 22 Slide © 2006 Thomson/South-Western n Using Tables of Binomial Probabilities Binomial Distribution

23. 23 Slide © 2006 Thomson/South-Western Binomial Distribution E ( x ) =  = np Var( x ) =  2 = np (1  p ) n Expected Value n Variance n Standard Deviation

24. 24 Slide © 2006 Thomson/South-Western Binomial Distribution E ( x ) =  = 3(.1) =.3 employees out of 3 Var( x ) =  2 = 3(.1)(.9) =.27 n Expected Value n Variance n Standard Deviation

25. 25 Slide © 2006 Thomson/South-Western A Poisson distributed random variable is often A Poisson distributed random variable is often useful in estimating the number of occurrences useful in estimating the number of occurrences over a specified interval of time or space over a specified interval of time or space A Poisson distributed random variable is often A Poisson distributed random variable is often useful in estimating the number of occurrences useful in estimating the number of occurrences over a specified interval of time or space over a specified interval of time or space It is a discrete random variable that may assume It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2,... ). an infinite sequence of values (x = 0, 1, 2,... ). It is a discrete random variable that may assume It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2,... ). an infinite sequence of values (x = 0, 1, 2,... ). Poisson Distribution

26. 26 Slide © 2006 Thomson/South-Western Examples of a Poisson distributed random variable: Examples of a Poisson distributed random variable: the number of knotholes in 14 linear feet of the number of knotholes in 14 linear feet of pine board pine board the number of knotholes in 14 linear feet of the number of knotholes in 14 linear feet of pine board pine board the number of vehicles arriving at a the number of vehicles arriving at a toll booth in one hour toll booth in one hour the number of vehicles arriving at a the number of vehicles arriving at a toll booth in one hour toll booth in one hour Poisson Distribution

27. 27 Slide © 2006 Thomson/South-Western Poisson Distribution n Two Properties of a Poisson Experiment 2. The occurrence or nonoccurrence in any interval is independent of the occurrence or interval is independent of the occurrence or nonoccurrence in any other interval. nonoccurrence in any other interval. 2. The occurrence or nonoccurrence in any interval is independent of the occurrence or interval is independent of the occurrence or nonoccurrence in any other interval. nonoccurrence in any other interval. 1. The probability of an occurrence is the same for any two intervals of equal length. for any two intervals of equal length. 1. The probability of an occurrence is the same for any two intervals of equal length. for any two intervals of equal length.

28. 28 Slide © 2006 Thomson/South-Western 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

29. 29 Slide © 2006 Thomson/South-Western Poisson Distribution MERCY n Example: Mercy Hospital Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening?

30. 30 Slide © 2006 Thomson/South-Western Poisson Distribution n Using the Poisson Probability Function  = 6/hour = 3/half-hour, x = 4

31. 31 Slide © 2006 Thomson/South-Western Poisson Distribution n Using Poisson Probability Tables

32. 32 Slide © 2006 Thomson/South-Western 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, …

33. 33 Slide © 2006 Thomson/South-Western Poisson Distribution A property of the Poisson distribution is that the mean and variance are equal.  A property of the Poisson distribution is that the mean and variance are equal.   =  2

34. 34 Slide © 2006 Thomson/South-Western Poisson Distribution n Variance for Number of Arrivals During 30-Minute Periods  =  2 = 3

35. 35 Slide © 2006 Thomson/South-Western Hypergeometric Distribution The hypergeometric distribution is closely related The hypergeometric distribution is closely related to the binomial distribution. to the binomial distribution. The hypergeometric distribution is closely related The hypergeometric distribution is closely related to the binomial distribution. to the binomial distribution. However, for the hypergeometric distribution: However, for the hypergeometric 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.

36. 36 Slide © 2006 Thomson/South-Western n Hypergeometric Probability Function 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

37. 37 Slide © 2006 Thomson/South-Western n Hypergeometric Probability Function Hypergeometric 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

38. 38 Slide © 2006 Thomson/South-Western Hypergeometric Distribution n Example: Neveready Bob Neveready has removed two dead batteries from a flashlight and inadvertently mingled them with the two good batteries he intended as replacements. The four batteries look identical. Bob now randomly selects two of the four batteries. What is the probability he selects the two good batteries? ZAP ZAP ZAP ZAP

39. 39 Slide © 2006 Thomson/South-Western Hypergeometric Distribution n Using the Hypergeometric Function where: x = 2 = number of good batteries selected x = 2 = number of good batteries selected n = 2 = number of batteries selected n = 2 = number of batteries selected N = 4 = number of batteries in total N = 4 = number of batteries in total r = 2 = number of good batteries in total r = 2 = number of good batteries in total

40. 40 Slide © 2006 Thomson/South-Western Hypergeometric Distribution n Mean n Variance

41. 41 Slide © 2006 Thomson/South-Western Hypergeometric Distribution n Mean n Variance

42. 42 Slide © 2006 Thomson/South-Western Hypergeometric Distribution Consider a hypergeometric 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. continued

43. 43 Slide © 2006 Thomson/South-Western Hypergeometric Distribution When the population size is large, a hypergeometric When the population size is large, a hypergeometric 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 hypergeometric When the population size is large, a hypergeometric 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 ).

44. 44 Slide © 2006 Thomson/South-Western End of Chapter 5