1 1 Slide © 2009 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University.

2 2 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Probability Distribution n Poisson Probability Distribution n Hypergeometric Probability Distribution

3 3 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Slide © 2009 Thomson South-Western. All Rights Reserved 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) n Example: JSL Appliances Discrete Random Variable with a Finite Number of Values We can count the TVs sold, and there is a finite We can count the TVs sold, and there is a finite upper limit on the number that might be sold (which is the number of TVs in stock).

5 5 Slide © 2009 Thomson South-Western. All Rights Reserved 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,... Discrete Random Variable with an Infinite Sequence of Values We can count the customers arriving, but there is We can count the customers arriving, but there is no finite upper limit on the number that might arrive. n Example: JSL Appliances

6 6 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Slide © 2009 Thomson South-Western. All Rights Reserved a tabular representation of the probability a tabular representation of the probability distribution for TV sales was developed. distribution for TV sales was developed. Using past data on TV sales, …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 n Example: JSL Appliances

10 Slide © 2009 Thomson South-Western. All Rights Reserved.10.20.30. 40.50 0 1 2 3 4 Values of Random Variable x (TV sales) ProbabilityProbability Discrete Probability Distributions n Example: JSL Appliances Graphicalrepresentation of probability distributionGraphicalrepresentation distribution

11 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Slide © 2009 Thomson South-Western. All Rights Reserved 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 n Example: JSL Appliances

14 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Variance n Example: JSL Appliances

17 Slide © 2009 Thomson South-Western. All Rights Reserved Binomial Probability 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

18 Slide © 2009 Thomson South-Western. All Rights Reserved Binomial Probability 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.

19 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Probability Distribution n Binomial Probability Function

20 Slide © 2009 Thomson South-Western. All Rights Reserved Binomial Probability 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

21 Slide © 2009 Thomson South-Western. All Rights Reserved Binomial Probability Distribution n Example: Evans Electronics Evans is concerned about a low retention rate for 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.

22 Slide © 2009 Thomson South-Western. All Rights Reserved Binomial Probability Distribution Let : p =.10, n = 3, x = 1 Choosing 3 hourly employees at random, what is Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year? n Example: Evans Electronics Using the probabilityfunction probabilityfunction

23 Slide © 2009 Thomson South-Western. All Rights Reserved Binomial Probability 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 n Example: Evans Electronics Using a tree diagram

29 Slide © 2009 Thomson South-Western. All Rights Reserved Binomial Probability Distribution E ( x ) =  = 3(.1) =.3 employees out of 3 Var( x ) =  2 = 3(.1)(.9) =.27 Expected Value Expected Value Variance Variance Standard Deviation Standard Deviation n Example: Evans Electronics

30 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Probability Distribution

31 Slide © 2009 Thomson South-Western. All Rights Reserved 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 Probability Distribution

32 Slide © 2009 Thomson South-Western. All Rights Reserved Poisson Probability 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.

33 Slide © 2009 Thomson South-Western. All Rights Reserved n Poisson Probability Function Poisson Probability 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 In Excel use EXP(1) e = 2.71828 In Excel use EXP(1)

34 Slide © 2009 Thomson South-Western. All Rights Reserved Poisson Probability Distribution n Example: Mercy Hospital Patients arrive at the emergency room of Mercy 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 What is the probability of 4 arrivals in 30 minutes on a weekend evening?

36 Slide © 2009 Thomson South-Western. All Rights Reserved Poisson Probability 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, … n Example: Mercy Hospital

37 Slide © 2009 Thomson South-Western. All Rights Reserved Poisson Probability 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

39 Poisson Probability Distributions  The limiting form of the binomial distribution where the probability of success is small and n is large is called the Poisson probability distribution  The binomial distribution becomes more skewed to the right (positive) as the probability of success become smaller  This distribution describes the number of times some event occurs during a specified interval (time, distance, area, or volume)  Examples of use:  Distribution of errors is data entry  Number of scratches in car panels

Poisson Probability Experiment 1.The Random Variable is the number of times (counting) some event occurs during a Defined Interval  Random Variable:  The Random Variable can assume an infinite number of values, however the probability becomes very small after the first few occurrences (successes)  Defined Interval:  The Defined Interval some kind of “Continuum” such as:  Misspelled words per page ( continuum = per page)  Calls per two hour period ( continuum = per two hour period)  Vehicles per day ( continuum = per day)  Goals per game ( continuum = per game)  Lost bags per flight ( continuum = per flight)  Defaults per 30 year mortgage period ( continuum = per 30 year mortgage period )  Interval = Continuum 2.The probability of the event is proportional to the size of the interval (the longer the interval, the larger the probability) 3.The intervals do not overlap and are independent (the number of occurrences in 1 interval does not affect the other intervals 4.When the probability of success is very small and n is large, the Poisson distribution is a limiting form of the binomial probability distribution (“Law of Improbable Events”) 40

Poisson Probability 41 In Excel use the POISSON function

42 Poisson Probability Distributions  Variance = Mu  Always positive skew (most of the successes are in the first few counts (0,1,2,3…)  As mu becomes larger, Poisson becomes more symmetrical  We can calculate Probability with only knowledge of Mu and X.  Although using the Binomial Probability Distribution is still technically correct, we can use the Poisson Probability Distribution to estimate the Binomial Probability Distribution when n is large and pi is small  Why? Because as n gets large and pi gets small, the Poisson and Binomial converge

43 Slide © 2009 Thomson South-Western. All Rights Reserved Hypergeometric Probability 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.

44 Slide © 2009 Thomson South-Western. All Rights Reserved n Hypergeometric Probability Function Hypergeometric Probability 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

45 Slide © 2009 Thomson South-Western. All Rights Reserved n Hypergeometric Probability Function Hypergeometric Probability Distribution for 0 < x < r number of ways x successes can be selected from a total of r successes in the population number of ways n – x failures can be selected from a total of N – r failures in the population number of ways a sample of size n can be selected from a population of size N

46 Slide © 2009 Thomson South-Western. All Rights Reserved Hypergeometric Probability Distribution Bob Neveready has removed two dead batteries 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. n Example: Neveready’s Batteries Bob now randomly selects two of the four Bob now randomly selects two of the four batteries. What is the probability he selects the two good batteries?

47 Slide © 2009 Thomson South-Western. All Rights Reserved Hypergeometric Probability Distribution 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 Using the probabilityfunction probabilityfunction n Example: Neveready’s Batteries

52 Slide © 2009 Thomson South-Western. All Rights Reserved Hypergeometric Probability 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. 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. 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 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. 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

53 Slide © 2009 Thomson South-Western. All Rights Reserved Hypergeometric Probability 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 ).

54 Hypergeometric Probability Distribution  Requirements for a Hypergeometric Experiment? 1.An outcome on each trial of an experiment is classified into 1 of 2 mutually exclusive categories: Success or Failure 2.The random variable is the number of counted successes in a fixed number of trials 3.The trials are not independent (For each new trial, the sample space changes 4.We assume that we sample from a finite population (number of items in population is known) without replacement and n/N > 0.05. So, the probability of a success changes for each trial  N = Number Of Items In Population  n = Number Of Items In Sample = Number Of Trials  S = Number Of Successes In Population  X = Number Of Successes In Sample

55 Hypergeometric Probability Distributio n  If the selected items are not returned to the population and n/N <0.05, then the Binomial Distribution can be used as a close approximation  In Excel use the HYPGEOMDIST function

1 1 Slide © 2009 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University.

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1 1 Slide © 2009 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University.

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