1. Probability distributions: part 1 BSAD 30 Dave Novak Source: Anderson et al., 2013 Quantitative Methods for Business 12 th edition – some slides are directly from J. Loucks © 2013 Cengage Learning

2. Covered so far… Chapter 1: Introduction What is modeling Types of models Basic problem formulation Review of basic linear (algebraic) problems Chapter 2: Introduction to probability Review of probability concepts (complement, union, intersection, conditional probability, joint probability table, independence, mutually exclusive) 2

3. Overview Random Variables Discrete Probability Distributions Uniform Probability Distribution Binomial Probability Distribution Poisson Probability Distribution Link to examples of types of discrete distributions http://www.epixanalytics.com/modelassist/AtRisk/ Model_Assist.htm#Distributions/Discrete_distribu tions/Discrete_distributions.htmhttp://www.epixanalytics.com/modelassist/AtRisk/ Model_Assist.htm#Distributions/Discrete_distribu tions/Discrete_distributions.htm 3

4. Overview We will briefly look at three “common” discrete probability examples Uniform Binomial Poisson In business applications, we often find instances of random variables that follow a discrete uniform, binomial, or Poisson probability distribution 4

5. What is a random variable? A random variable (RV) is a numerical description of the outcome of an experiment Keep in mind that there is a difference between numeric variables and categorical variables Numeric: temperature, speed, age, monetized data, etc. Categorical: state of residence, gender, blood type, etc. 5

6. What is a random variable? Two types of random variables: Discrete Continuous 6

7. Random variables 7

8. Question Random Variable x Type Familysize x = Number of dependents in family reported on tax return Discrete Distance from home to store x = Distance in miles from 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) 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 Discrete 8

9. Example Discrete random variable (RV) with a finite number of possible values There is a readily identifiable upper bound to the number of TVs sold on any given day In this case, no more than 4 TVs sold Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) 9

10. Example Discrete random variable (RV) with an infinite number of possible values There is no readily identifiable upper bound on the number of customers coming into the store on any given day There cannot be an infinite # of customers, but we are not setting an upper bound (could be 75, 500, or 2,000) 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,... 10

11. Discrete probability distributions The probability distribution for a random variable describes how probabilities associated with each value are distributed (or allocated) over all possible values We can describe a discrete probability distribution with a table, graph, or equation In the TV sales example, we would want a mathematical and/or visual representation of the probability of selling 0, 1, 2, 3, or 4 TVs on any given day 11

12. Discrete probability distributions The probability distribution is defined by a probability function, denoted by f(x), which provides the probability for each value of the random variable The function f(x) is a mathematical representation of the probability distribution The following conditions are required: f(x) > 0  f(x) = 1 12

13. Discrete distribution: DiCarlo motors example Using historical data on car sales, a tabular representation of sales is created Number Number Units Sold of Days Units Sold of Days 0 54 1 117 1 117 2 72 2 72 3 42 3 42 4 12 4 12 5 3 300 300 x f(x) x f(x) 0.18 0.18 1.39 1.39 2.24 2.24 3.14 3.14 4.04 4.04 5.01 5.01 1.00 1.00 x f(x) x f(x) 0.18 0.18 1.39 1.39 2.24 2.24 3.14 3.14 4.04 4.04 5.01 5.01 1.00 1.00.18 = 54/300.04 = 12/300 13

14. Discrete distribution: DiCarlo motors example Graphical representation.10.20.30. 40.50 0 1 2 3 4 5 Values of Random Variable x (car sales) ProbabilityProbability 14

15. Discrete distribution: DiCarlo motors example The probability distribution provides the following information There is a 0.18 probability that no cars will be sold during a day  f(0) = 18% The most probable sales volume is 1, with f(1) = 0.39  f(1) = 39% There is a 0.05 probability of either four or five cars being sold  f(4) + f(5) = 5% 15

16. Summary Up to this point, we have not discussed the specific TYPE of discrete probability distribution (i.e. uniform, binomial, Poisson, etc.) We have only discussed probability distributions in terms of being discrete as opposed to continuous A review of basic statistical concepts is next 16

17. Expected value and variance The expected value, or mean, of a random variable is a measure of its central location Mean, median, and mode are measures of central tendency because they identify a single value as “typical” or representative of all values in a probability distribution E(x) =  =  x f(x) 17

18. Expected value and variance The variance,  , summarizes the variability in the values of a random variable The standard deviation, , is defined as the positive square root of the variance Var(x) =  2 =  (x -  ) 2 f(x) 18

19. Expected value and variance Both the StdDev and variance provide a measure of how much the values in the probability distribution differ from the mean The higher the standard deviation, the more different the different observations are from one another and from the mean When a probability distribution has a high standard deviation, the mean is not a good measure of central tendency 19

20. Expected value and variance Scores = 1,4,3,4,2,7,18,3,7,2,4,3 Mean = 5 Median = 3.5 Standard Deviation = 4.53 The standard deviation indicates that the average difference between each score and the mean is around 4.5 points. However, only one score (18) is 4.5 or more points different from the mean. The one extreme score (18) overly influences the mean. The median (3.5) is a better measure of central tendency in this case because extreme scores do not influence the median 20

21. Discrete distribution: DiCarlo motors example Number Number Units Sold of Days Units Sold of Days 0 54 1 117 1 117 2 72 2 72 3 42 3 42 4 12 4 12 5 3 300 300 x f(x) x f(x) 0.18 0.18 1.39 1.39 2.24 2.24 3.14 3.14 4.04 4.04 5.01 5.01 1.00 1.00 x f(x) x f(x) 0.18 0.18 1.39 1.39 2.24 2.24 3.14 3.14 4.04 4.04 5.01 5.01 1.00 1.00 21

22. DiCarlo motors example Calculate expected value of discrete RV expected number of cars sold in a day x f(x) xf(x) x f(x) xf(x) 0.18.00 0.18.00 1.39.39 1.39.39 2.24.48 2.24.48 3.14.42 3.14.42 4.04.16 4.04.16 5.01.05 5.01.05 E(x) = 1.50 E(x) = 1.50 22 0 x 0.18 = 0 1 x 0.39 = 0.39

23. DiCarlo motors example Calculate variance and StdDev 012345 -1.5-0.5 0.5 0.5 1.5 1.5 2.5 2.5 3.5 3.5 2.25 2.25 0.25 0.25 2.25 2.25 6.25 6.2512.25.18.39.24.14.04.01.4050.0975.0600.3150.2500.1225 x -  (x -  ) 2 f(x)f(x)f(x)f(x) (x -  ) 2 f(x) Variance of daily sales =  2 = 1.2500 x carssquaredcarssquared 23

24. DiCarlo motors example Calculate variance and StdDev 24 Var(x) =  2 =  (x -  ) 2 f(x) = 0.4050 + 0.0975 + 0.0600 + 0.3150 + 0.2500 + 0.1225 Var(x) =  2 = 1.25 Var(x) =  2 =  (x -  ) 2 f(x) = 0.4050 + 0.0975 + 0.0600 + 0.3150 + 0.2500 + 0.1225 Var(x) =  2 = 1.25

25. Expected value and variance From a decision-making or analyst perspective what are some of the practical implications of this discussion? If the data you are analyzing have a high variance, making decisions based on the mean, or even stressing the importance of the average, is likely to be misleading 25

26. Expected value and variance What should you do? Generate a visual representation of the data! You need to better characterize the data to see if they fit into any well-known families of probability distributions – this would be the first step in analysis Knowing what the data “aren’t” is also useful 26

27. Expected value and variance What should you do? Knowing that data do not follow a particular distribution is important in terms of analysis There are particular characteristics associated with different types of distributions that can guide you in your analysis 27

28. Discrete Distributions we will examine 1) Uniform 2) Binomial or Bernoulli 3) Poisson 28

29. Discrete uniform probability distribution The discrete uniform probability distribution is the simplest example of a discrete probability distribution given by a formula Example: getting a 1, 2, 3, 4, 5, or 6 when rolling single die – f(x) = 1/6 f(x) = 1/n where: n = the number of values the random variable may assume the values of the random variable random variable are equally likely are equally likely the values of the random variable random variable are equally likely are equally likely 29

30. Binomial probability distribution Also known as Bernoulli distribution Has four properties: 1) Experiment consists of n, independent trials 2) Only TWO outcomes are possible for each trial (success/failure, good/bad, on/off, yes/no, etc.) 3) The probability of success stays the same for all trials 4) All trials are independent 30

31. Binomial probability distribution We are interested in the number of successes, or positive outcomes occurring in the n trials x denotes the number of successes, or positive outcomes occurring in the n trials where: f(x) = the probability of x successes in n trials n = the number of trials p = the probability of success on any one trial 31

32. Binomial probability distribution 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 32

33. Binomial probability distribution Assume the probability that any customer who comes into a store and actually makes a purchase is 0.3 (30% chance of success) What is the probability that 2 of the next 3 customers who enter the store make a purchase? Identify: n, x, p 33

34. Binomial probability distribution 34

35. Binomial probability distribution (decision tree) 1 st Customer 2 nd Customer 3 rd Customer x x Prob. Purchases (.3) Purchases (.3) (.7) Does Not Purchase (.7) Does Not Purchase 3 3 2 2 0 0 2 2 2 2 Purchases (.3) Purchases (.3) Purchases (.3) Purchases (.3) DNP (.7) Does Not Purchase (.7) Does Not Purchase (.7) Does Not Purchase (.7) Does Not Purchase (.7) DNP (.7) P (.3).027.063.343.063 1 1 1 1.147 11 35

36. Binomial probability distribution If a six-sided die is rolled three times, what is the probability that the number 5 comes up twice? Identify: n, x, p 36

37. Binomial probability distribution 37

38. Binomial probability distribution 1 st roll 2 nd roll 3 rd roll x x Prob. Success “5” (.17) Success “5” (.17) (.83) Failure (1,2, 3, 4, 6) (.83) Failure (1,2, 3, 4, 6) 3 3 2 2 0 0 2 2 2 2 Success (.17) Success (.17) F (.83) Failure (.83) S (.17).005.572.024 1 1 1 1.117 11 Success (.17) Success (.17) Failure (.83) S (.17) F (.83) 38.024.117

39. Binomial probability distribution What’s the probability if I roll a die 10 times, the number 5 comes up four times? Identify: n, x, p 39

40. Binomial probability distribution Expected value Variance Standard deviation E(x) =  = np Var(x) =  2 = np(1  p) 40

41. Binomial probability distribution In the clothing store example, calculate: Expected value Variance Standard deviation 41

42. Poisson probability distribution A Poisson distributed random variable is often useful in estimating the number of occurrences over a specified interval of time or space which can be counted in whole numbers Very useful in RISK analysis It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2,... ∞) 42

43. Poisson probability distribution How is an RV that follows a Poisson distribution different from an RV that follows a binomial distribution? It is possible to count how many events have occurred, but meaningless to ask how many events have NOT occurred In the binomial situation, we know the probability of two mutually exclusive events (p, q) – in the Poisson situation, we have no q (it has only one parameter the average frequency an event occurs) 43

44. Poisson probability distribution Examples Number of customers arriving at a supermarket checkout between 5 PM and 6 PM Number of text messages you receive over the course of a week Number of car accidents over the course of a year 44

45. Poisson probability distribution Two properties of Poisson distributions 1) The probability of occurrence is the same over any two time intervals of equal length 2) The occurrence or nonoccurrence in any time interval is independent of occurrence or nonoccurrence in any other time interval 45

46. Poisson probability distribution where: f(x) = probability of x occurrences in an interval = mean number of occurrences in an interval e = 2.71828 46 For more info: https://en.wikipedia.org/wiki/E_(mathematical_constant)

47. Drive-up teller window example Suppose that we are interested in the number of cars arriving at the drive-up teller window of a bank during a 15-minute period on weekday mornings We assume that the probability of a car arriving is the same for any two time periods of equal length (i.e. prob of a car arriving in the first minute is exactly the same as the prob of a car arriving in the last minute), and the arrival or non-arrival of a car in any time period is independent of the arrival or non-arrival in any other time period An analysis of historical data shows that the average number of cars arriving during a 15-minute interval of time is 10, so the Poisson probability function with = 10 applies 47

48. Drive-up teller window example = 10 arrivals / 15 minutes, x = 5 We want to know the probability that exactly 5 cars will arrive over the 15 minute time interval Identify: x and X = 5 => we are given that there are 10 arrivals every 15 minutes, so the average # of arrivals over the time period is 10 48

49. Drive-up teller window example = 10 arrivals / 15 minutes, x = 5 So, there is a 3.78% chance that exactly 5 cars will arrive over the 15 minute time period 49

50. Highway defect example Suppose that we are concerned with the occurrence of major defects in a section of highway one month after that section was resurfaced We assume that the probability of a defect is the same for any two highway intervals of equal length (i.e. the probability of a defect between mile markers 1 and 2 is the same as the probability of a defect between mile markers 4 and 5, etc.) and that the occurrence of a defect in any one mile interval is independent of the occurrence or nonoccurrence of a defect in any other interval Thus, the Poisson probability distribution applies 50

51. Highway defect example Find the probability that no major defects occur in a specific 3-mile stretch of highway assuming that major defects occur at the average rate of two defects per mile 51

52. Highway defect example 52

53. Poisson probability distribution Expected value Variance Standard deviation E(x) = µ =  the rate or frequency of an event Var(x) =  2 = Var(x) =  2 = 53

54. Highway defect example In the highway defect example, calculate: Expected value Variance Standard deviation 54

55. Summary Discussion of random variables Discrete Continuous Examples of discrete probability distributions Uniform Binomial Poisson 55