Discrete probability s BSAD 30 Dave Novak Fall 08 Source: Anderson et al., 05 Quantitative Methods for Business th edition some slides are directly from J. Loucks 03 Cengage Learning Covered so far Chapter : Introduction What is modeling? 3 types of models Basic problem formulation Review of basic linear (algebraic) problems Chapter : Introduction to probability Review of probability concepts (complement, union, intersection, conditional probability, joint probability table, independence, mutually exclusive) Overview discrete s Random Variables Discrete Probability Distributions Uniform Probability Distribution Binomial Probability Distribution Poisson Probability Distribution Link to examples of types of discrete s http://www.epixanalytics.com/modelassist/atrisk/ Model_Assist.htm#Distributions/Discrete_distribu tions/discrete_s.htm 3 Overview discrete s What is a probability? 4 A way to view relationship between a particular outcome and the probability that particular outcomes occurs A table, equation, or graphical representation that links the possible outcomes of an experiment to their likelihood (probability) of occurrence Overview discrete s We will briefly look at three common discrete probability examples Uniform Binomial Poisson In business applications, we often find instances of discrete random variables that follow a uniform, binomial, or Poisson probability 5 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. 6
Random variable? Random variables Two types of numeric random variables: Discrete Continuous 7 8 Random variables Question Random Variable x Type Family size Distance from home to store Own dog or cat 9 x = Number of dependents in family reported on tax return x = Distance in miles from home to the store site x = if own no pet; = if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s) Discrete Continuous Discrete Example Discrete random variable (RV) with a finite number of possible values 0 Let x = number of TVs sold at the store in one day, where x can take on 5 values (0,,, 3, 4) There are a limited number of values that x can take on There is an identifiable upper bound and lower bound to the number of TVs sold on any given day In this case, no fewer than 0 and no more than 4 TVs are sold Example Discrete random variable (RV) with a finite number of possible values Let x = the grade a student receives in a particular class where x can take on 3 values (A+, A, A-, B+, B, B-, C+ C, C-, D+, D, D-, F) There are a limited number of possible outcomes for x Example Discrete random variable (RV) with an infinite number of possible values Let x = number of customers arriving in one day, where x can take on the values 0,,,... 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,000)
Probability 8/3/08 Discrete probability s The probability for a RV describes how probabilities associated with each value are distributed (or allocated) over all possible values We can describe a discrete probability 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,,, 3, or 4 TVs on any given day 3 Discrete probability s The probability 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 The following conditions are required: 4 f(x) > 0 f(x) = Sample Space = { 0,,, 3, 4 } Discrete : Using historical data on car sales, a tabular 5 representation of sales is created Number Units Sold of Days 0 54 7 7 3 4 4 5 3 300 x f(x) 0.8.39.4 3.4 4.04 5.0.00 f(0) = 0.8 = 54/300 f(4) = 0.04 = /300 Discrete : Graphical representation 6.50.40.30.0.0 0 3 4 5 Values of Random Variable x (car sales) Discrete : The probability provides the following information There is a 0.8 probability that no cars will be sold during a day f(0) = 8% The most probable sales volume is, with f() = 0.39 f() = 39% There is a 0.05 probability of either four or five cars being sold f(4) + f(5) = 5% Up to this point, we have not discussed the specific TYPE of discrete probability (i.e. uniform, binomial, Poisson, etc.) We have only discussed RVs in terms of being discrete as opposed to continuous A review of basic statistical concepts is next 7 8 3
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 9 E(x) = = x f(x) The,, summarizes the variability in the values of a random variable The standard deviation,, is defined as the positive square root of the 0 Var(x) = = (x - ) f(x) StdDev(x) = = Both the StdDev and provide a measure of how much the values in the probability 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 has a high standard deviation, the mean is generally not a good measure of central tendency Scores =,4,3,4,,7,8,3,7,,4,3 Mean = 58/ = 4.83 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 (8) is more than SD above the mean. The one extreme score (8) 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 Discrete : 3 Number Units Sold of Days 0 54 7 7 3 4 4 5 3 300 x f(x) 0.8.39.4 3.4 4.04 5.0.00 Calculate expected value of discrete RV 4 x f(x) xf(x) 0.8.00.39.39.4.48 3.4.4 4.04.6 5.0.05 E(x) =.50 expected number of cars sold in a day 4
Calculate and StdDev Calculate and StdDev x 0 3 4 5 x - (x - ) f(x) (x - ) f(x) -.5-0.5 0.5.5.5 3.5.5 0.5 0.5.5 6.5.5.8.39.4.4.04.0 Variance of daily sales = =.500.4050.0975.0600.350.500.5 cars squared Var(x) = = (x - ) f(x) = 0.4050 + 0.0975 + 0.0600 + 0.350 + 0.500 + 0.5 Var(x) = =.5 Standard deviation of daily sales = =.500 =.8 cars Standard deviation of daily sales = =.500 =.8 cars 5 6 From a decision-making perspective what are some of the practical implications? If the data you are analyzing have a high, making decisions based on the mean, or even stressing the importance of the average, is likely to be misleading The median might be a better measure of central tendency What should you do? Generate a visual representation of the data! You need to characterize the data to see if they fit into any well-known families of probability s this would be the first step in analysis Are data skewed or symmetrical? Knowing what the data aren t is also useful 7 8 Visual representation What should you do? Knowing that data do not follow a particular is important in terms of analysis There are particular characteristics associated with different types of s that can guide you in your analysis 9 30 5
Discrete Distributions we will examine ) Uniform ) Binomial or Bernoulli 3) Poisson 3 Discrete uniform probability The discrete uniform probability is the simplest example of a discrete probability given by a formula 3 the values of the f(x) = /n random variable are equally likely where: n = the number of values the random variable may assume Example: getting a,, 3, 4, 5, or 6 when rolling single die f(x) = /6 or 6.7% Discrete uniform probability If grade outcomes were distributed uniformly, then: getting an A+, A, A-, B+, F would be equally likely and f(x) = /3 or 7.7% 33 the values of the f(x) = /n random variable are equally likely where: n = the number of values the random variable may assume (3 in this case) f(a+) = 0.077, f(a) = 0.077, etc. Also known as Bernoulli Has four properties: ) Experiment consists of n, independent trials ) 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 34 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 35 n! x ( n x) f ( x) p ( p) x!( n x)! 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 n! x f ( x) p ( p) x!( n x)! Number of experimental outcomes providing exactly x successes in n trials 36 ( n x) Probability of a particular sequence of trial outcomes with x successes in n trials 6
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 of the next 3 customers who enter the store make a purchase? Identify: n, x, p 37 38 (decision tree) 39 st Customer nd Customer 3 rd Customer x Prob. Purchases P (.3) 3.07 (.3) Purchases (.3) DNP (.7) P (.3).063.063 Does Not Purchase (.7) DNP (.7).47 (.7) Does Not Purchase Purchases (.3) Does Not Purchase (.7) P (.3) DNP (.7) P (.3) DNP (.7) 0.063.47.47.343 If a six-sided die is rolled three times, what is the probability that the number 5 comes up once? Identify: n, x, p 40 4 4 st roll nd roll 3 rd roll x Prob. Success S (.7) 3.005 (.7) Success 5 F (.83).04 (.7) S (.7).04 Failure (.83) F (.83).7 (.83) Failure (,, 3, 4, 6) Success (.7) Failure (.83) S (.7) F (.83) S (.7) F (.83) 0.04.7.7.57 7
What s the probability that if flip a coin 0 times, I get a Heads exactly 7 times? Identify: n, x, p What s the probability that if flip a coin 0 times, I get a Heads at least 8 times? Identify: n, x, p 43 44 Expected value E(x) = = np Variance Var(x) = = np( p) Standard deviation 45 np( p) In the clothing store example, calculate: 46 Expected value E(x) = = np = 3(0.3) = 0.9 Variance Var(x) = = np( p) = 3(0.3)(0.7) = 0.63 Standard deviation np( p) = 0.794 A Poisson distributed RV 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 RV that may assume an infinite sequence of values (x = 0,,, 3, 4, 5.. ) 47 Poisson versus Binomial How is an RV that follows a Poisson different from an RV that follows a binomial? With Poisson, we are counting the number of occurrences (x) in a particular interval, given an average rate of occurrence in that interval (λ) With Binomial, we are counting the number of occurrences (x), given a fixed number of possibilities or trials (n), where a single 48 occurrence happens with probability (p) 8
Poisson versus Binomial Binomial Distribution Fixed Number of Trials (n) [0 pie throws] Only Possible Outcomes [hit or miss] Probability of Success is Constant (p) [0.4 success rate] Each Trial is Independent [throw has no effect on throw ] Poisson Distribution Infinite Number of Trials Unlimited Number of Outcomes Possible Mean of the Distribution is the Same for All Intervals (mu) Number of Occurrences in Any Given Interval Independent of Others Poisson versus Binomial In the Binomial situation, we know the probability of two mutually exclusive events (p, q), where p = q In the Poisson situation, we only one parameter, λ, which is the average frequency that an event occurs in a particular interval of time or distance, etc. Predicts Number of Successes within a Set Number of Trials Predicts Number of Occurrences per Unit Time, Space,... 49 Source: http://www.cedarcrest.edu/academic/bio/hale/biostat/sessionlinks/bvsp.html 50 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 Two properties of Poisson s ) The probability of occurrence is the exactly the same over any two time intervals of equal length If the eruption of a volcano follows a Poisson, then the probability the volcano erupts this year is the same as the probability that it erupts next year, and in all subsequent years 5 5 Two properties of Poisson s ) The occurrence or nonoccurrence in any time interval is independent of occurrence or nonoccurrence in any other time interval If a major flood event happens this year (a very rare occurrence), the chance that it happens again next year is INDEPENDENT of the fact that it happened this year In Vermont we have had several major flood events ( 00-year events) over the past 5 years 53 x e f ( x) x! where: f(x) = probability of x occurrences in an interval = mean number of occurrences in an interval e =.788 54 For more info: https://en.wikipedia.org/wiki/e_(mathematical_constant) 9
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 5-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 5-minute interval of time is 0, so the function with = 0 applies 55 Drive-up teller window example We want to know the probability that exactly 5 cars will arrive over the 5 minute time interval Identify: x and = 0 arrivals / 5 minutes, x = 5 X = 5 => we are given that there are 0 arrivals every 5 minutes, so the average # of arrivals over the time period is 0 56 Drive-up teller window example Drive-up teller window example 57 = 0 arrivals / 5 minutes, x = 5 5 0 0 (.788) f (5).0378 5! So, there is a 3.78% chance that exactly 5 cars will arrive over the 5 minute time period Assume that we want to know the probability that AT LEAST car will arrive over a ONE minute time interval = 0 arrivals / 5 minutes, x = 5 Identify: x and X (where upper bound is presumably limitless) => we are given that there are 0 arrivals every 5 minutes, so the average # of arrivals over a oneminute time period is 0/5 = 0.667 arrivals / minute 58 Drive-up teller window example 59 = 0 arrivals / 5 minutes, x = 5 Highway defect example 60 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 and is the same as the probability of a defect between mile markers 3, 3 4, 4 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 applies 0
Highway defect example 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 6 6 Expected value E(x) = µ = the rate or frequency of an event Variance Var(x) = = Standard deviation 63 = Highway defect example In the highway defect example, calculate: Expected value E(x) = µ = = Variance Var(x) = = Standard deviation 64 = Summary Random Variables Discrete Continuous Review of measures of central tendancy Discrete Probability Distributions Uniform Probability Distribution Binomial Probability Distribution Poisson Probability Distribution 65