1. 1 1 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. SLIDES. BY John Loucks St. Edward’s University......................

2. 2 2 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 5 Discrete Probability Distributions.10.20.30.40 0 1 2 3 4 n Random Variables n Developing Discrete Probability Distributions n Expected Value and Variance n Binomial Probability Distribution Distribution n Poisson Probability Distribution Distribution n Hypergeometric Probability Distribution Distribution n Bivariate Distributions, Covariance, and Financial Portfolios and Financial Portfolios

3. 3 3 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 5 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 6 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 formula. with a table, graph, or formula. We can describe a discrete probability distribution We can describe a discrete probability distribution with a table, graph, or formula. with a table, graph, or formula. Discrete Probability Distributions

8. 8 8 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Two types of discrete probability distributions will Two types of discrete probability distributions will be introduced. be introduced. Two types of discrete probability distributions will Two types of discrete probability distributions will be introduced. be introduced. First type: uses the rules of assigning probabilities First type: uses the rules of assigning probabilities to experimental outcomes to determine probabilities to experimental outcomes to determine probabilities for each value of the random variable. for each value of the random variable. First type: uses the rules of assigning probabilities First type: uses the rules of assigning probabilities to experimental outcomes to determine probabilities to experimental outcomes to determine probabilities for each value of the random variable. for each value of the random variable. Discrete Probability Distributions Second type: uses a special mathematical formula Second type: uses a special mathematical formula to compute the probabilities for each value of the to compute the probabilities for each value of the random variable. random variable. Second type: uses a special mathematical formula Second type: uses a special mathematical formula to compute the probabilities for each value of the to compute the probabilities for each value of the random variable. random variable.

9. 9 9 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The probability distribution is defined by a The probability distribution is defined by a probability function, denoted by f ( x ), that provides probability function, denoted by f ( x ), that 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 ), that provides probability function, denoted by f ( x ), that 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

10. 10 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Discrete Probability Distributions There are three methods for assign probabilities to There are three methods for assign probabilities to random variables: classical method, subjective random variables: classical method, subjective method, and relative frequency method. method, and relative frequency method. There are three methods for assign probabilities to There are three methods for assign probabilities to random variables: classical method, subjective random variables: classical method, subjective method, and relative frequency method. method, and relative frequency method. The use of the relative frequency method to develop The use of the relative frequency method to develop discrete probability distributions leads to what is discrete probability distributions leads to what is called an empirical discrete distribution. called an empirical discrete distribution. The use of the relative frequency method to develop The use of the relative frequency method to develop discrete probability distributions leads to what is discrete probability distributions leads to what is called an empirical discrete distribution. called an empirical discrete distribution. example on next slide example on next slide

11. 11 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 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 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 80/20080/200 Discrete Probability Distributions n Example: JSL Appliances

12. 12 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part..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

13. 13 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Discrete Probability Distributions In addition to tables and graphs, a formula that In addition to tables and graphs, a formula that gives the probability function, f ( x ), for every value gives the probability function, f ( x ), for every value of x is often used to describe the probability of x is often used to describe the probability distributions. distributions. In addition to tables and graphs, a formula that In addition to tables and graphs, a formula that gives the probability function, f ( x ), for every value gives the probability function, f ( x ), for every value of x is often used to describe the probability of x is often used to describe the probability distributions. distributions. Several discrete probability distributions specified Several discrete probability distributions specified by formulas are the discrete-uniform, binomial, by formulas are the discrete-uniform, binomial, Poisson, and hypergeometric distributions. Poisson, and hypergeometric distributions. Several discrete probability distributions specified Several discrete probability distributions specified by formulas are the discrete-uniform, binomial, by formulas are the discrete-uniform, binomial, Poisson, and hypergeometric distributions. Poisson, and hypergeometric distributions.

14. 14 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 the values of the random variable random variable are equally likely are equally likely

15. 15 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Expected Value 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 expected value is a weighted average of the The expected value is a weighted average of the values the random variable may assume. The values the random variable may assume. The weights are the probabilities. weights are the probabilities. The expected value is a weighted average of the The expected value is a weighted average of the values the random variable may assume. The values the random variable may assume. The weights are the probabilities. weights are the probabilities. The expected value does not have to be a value the The expected value does not have to be a value the random variable can assume. random variable can assume. The expected value does not have to be a value the The expected value does not have to be a value the random variable can assume. random variable can assume. E ( x ) =  =  xf ( x )

16. 16 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Variance and Standard Deviation 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 variance is a weighted average of the squared The variance is a weighted average of the squared deviations of a random variable from its mean. deviations of a random variable from its mean. The weights are the probabilities. The weights are the probabilities. The variance is a weighted average of the squared The variance is a weighted average of the squared deviations of a random variable from its mean. deviations of a random variable from its mean. The weights are the probabilities. The weights are the probabilities. Var ( x ) =  2 =  ( x -  ) 2 f ( x ) The standard deviation, , is defined as the The standard deviation, , is defined as the positive square root of the variance. positive square root of the variance. The standard deviation, , is defined as the The standard deviation, , is defined as the positive square root of the variance. positive square root of the variance.

17. 17 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

18. 18 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 TVssquaredTVssquared Standard deviation of daily sales = 1.2884 TVs Variance n Example: JSL Appliances

19. 19 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Bivariate Distributions A probability distribution involving two random A probability distribution involving two random variables is called a bivariate probability distribution. variables is called a bivariate probability distribution. A probability distribution involving two random A probability distribution involving two random variables is called a bivariate probability distribution. variables is called a bivariate probability distribution. Each outcome of a bivariate experiment consists of Each outcome of a bivariate experiment consists of two values, one for each random variable. two values, one for each random variable. Each outcome of a bivariate experiment consists of Each outcome of a bivariate experiment consists of two values, one for each random variable. two values, one for each random variable. When dealing with bivariate probability distributions, When dealing with bivariate probability distributions, we are often interested in the relationship between we are often interested in the relationship between the random variables. the random variables. When dealing with bivariate probability distributions, When dealing with bivariate probability distributions, we are often interested in the relationship between we are often interested in the relationship between the random variables. the random variables. Example: rolling a pair of dice

20. 20 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. A company asked 200 of its employees how they rated their benefit package and job satisfaction. The rated their benefit package and job satisfaction. The crosstabulation below shows the ratings data. crosstabulation below shows the ratings data. A Bivariate Discrete Probability Distribution Benefits Package ( x ) 1 2 3 Total 123 28 26 4 28 26 4 58 58 98 98 44 44 52 78 70 52 78 70 Total 200 22 42 34 22 42 34 Job Satisfaction ( y ) Job Satisfaction ( y ) 2 10 32 2 10 32

21. 21 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The bivariate empirical discrete probabilities for benefits rating and job satisfaction are shown below. benefits rating and job satisfaction are shown below. Benefits Package ( x ) 1 2 3 Total 123.14.13.02.14.13.02.29.49.22.26.39.35 Total 1.00.11.21.17.11.21.17 Job Satisfaction ( y ) Job Satisfaction ( y ).01.05.16.01.05.16 A Bivariate Discrete Probability Distribution

22. 22 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. x f(x)f(x) xf ( x ) x - E ( x )( x - E ( x )) 2 ( x - E ( x )) 2 f ( x ) 10.29 -0.930.86490.250821 20.490.98 0.070.00490.002401 30.220.661.071.14490.251878 E ( x ) =1.93 Var ( x ) =0.505100 A Bivariate Discrete Probability Distribution n Expected Value and Variance for Benefits Package, x

23. 23 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. y f(y)f(y) yf ( y ) y - E ( y )( y - E ( y )) 2 ( y - E ( y )) 2 f ( y ) 10.26 -1.091.18810.308906 20.390.78-0.090.00810.003159 30.351.05 0.910.82810.289835 E ( y ) =2.09 Var ( y ) =0.601900 A Bivariate Discrete Probability Distribution n Expected Value and Variance for Job Satisfaction, y

24. 24 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. s f(s)f(s) sf ( s ) s - E ( s )( s - E ( s )) 2 ( s - E ( s )) 2 f ( s ) 20.140.28-2.024.08040.571256 30.240.72-1.021.04040.249696 40.240.96-0.020.00040.000960 50.221.100.980.96040.211376 60.160.961.983.92040.627264 E(s) =E(s) =4.02 Var ( s ) =1.660552 Expected Value and Variance for Bivariate Distrib. Expected Value and Variance for Bivariate Distrib. A Bivariate Discrete Probability Distribution

25. 25 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Var xy = [ Var ( x + y ) – Var ( x ) – Var ( y )]/2 Var xy = [1.660552 – 0.5051 – 0.6019]/2 = 0.276776 A Bivariate Discrete Probability Distribution Covariance for Random Variables x and y Covariance for Random Variables x and y

26. 26 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Correlation Between Variables x and y Correlation Between Variables x and y A Bivariate Discrete Probability Distribution

27. 27 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. stationarityassumptionstationarityassumption

28. 28 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

29. 29 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. where: where: x = the number of successes x = the number of successes p = the probability of a success on one trial p = the probability of a success on one trial n = the number of trials n = the number of trials f ( x ) = the probability of x successes in n trials f ( x ) = the probability of x successes in n trials n ! = n ( n – 1)( n – 2) ….. (2)(1) n ! = n ( n – 1)( n – 2) ….. (2)(1) Binomial Probability Distribution n Binomial Probability Function

30. 30 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

31. 31 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Binomial Probability Distribution n Example: Evans Electronics Evans Electronics is concerned about a low Evans Electronics is concerned about a low retention rate for its employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. 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? Thus, for any hourly employee chosen at random, 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.

32. 32 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Binomial Probability Distribution n Example: Evans Electronics The probability of the first employee leaving and the second and third employees staying, denoted ( S, F, F ), is given by p (1 – p )(1 – p ) With a.10 probability of an employee leaving on any one trial, the probability of an employee leaving on the first trial and not on the second and third trials is given by (.10)(.90)(.90) = (.10)(.90) 2 =.081 (.10)(.90)(.90) = (.10)(.90) 2 =.081

33. 33 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Binomial Probability Distribution n Example: Evans Electronics Two other experimental outcomes also result in one success and two failures. The probabilities for all three experimental outcomes involving one success follow. ExperimentalOutcome ( S, F, F ) ( F, S, F ) ( F, F, S ) Probability of Experimental Outcome p (1 – p )(1 – p ) = (.1)(.9)(.9) =.081 (1 – p ) p (1 – p ) = (.9)(.1)(.9) =.081 (1 – p )(1 – p ) p = (.9)(.9)(.1) =.081 Total =.243

34. 34 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Binomial Probability Distribution Let : p =.10, n = 3, x = 1 n Example: Evans Electronics Using the probabilityfunction probabilityfunction

35. 35 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

36. 36 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Binomial Probabilities and Cumulative Probabilities With modern calculators and the capability of With modern calculators and the capability of statistical software packages, such tables are statistical software packages, such tables are almost unnecessary. almost unnecessary. With modern calculators and the capability of With modern calculators and the capability of statistical software packages, such tables are statistical software packages, such tables are almost unnecessary. almost unnecessary. These tables can be found in some statistics These tables can be found in some statistics textbooks. textbooks. These tables can be found in some statistics These tables can be found in some statistics textbooks. textbooks. Statisticians have developed tables that give Statisticians have developed tables that give probabilities and cumulative probabilities for a probabilities and cumulative probabilities for a binomial random variable. binomial random variable. Statisticians have developed tables that give Statisticians have developed tables that give probabilities and cumulative probabilities for a probabilities and cumulative probabilities for a binomial random variable. binomial random variable.

37. 37 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Using Tables of Binomial Probabilities Binomial Probability Distribution

38. 38 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Binomial Probability Distribution E ( x ) =  = np Var ( x ) =  2 = np (1  p ) n Expected Value n Variance n Standard Deviation

39. 39 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Binomial Probability Distribution E ( x ) = np = 3(.1) =.3 employees out of 3 Var ( x ) = np (1 – p ) = 3(.1)(.9) =.27 Expected Value Expected Value Variance Variance Standard Deviation Standard Deviation n Example: Evans Electronics

40. 40 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

41. 41 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Examples of Poisson distributed random variables: Examples of Poisson distributed random variables: 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 toll the number of vehicles arriving at a toll booth in one hour booth in one hour the number of vehicles arriving at a toll the number of vehicles arriving at a toll booth in one hour booth in one hour Poisson Probability Distribution Bell Labs used the Poisson distribution to model Bell Labs used the Poisson distribution to model the arrival of phone calls. the arrival of phone calls. Bell Labs used the Poisson distribution to model Bell Labs used the Poisson distribution to model the arrival of phone calls. the arrival of phone calls.

42. 42 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

43. 43 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Poisson Probability Function Poisson Probability Distribution where: where: x = the number of occurrences in an interval x = the number of occurrences in an interval f(x) = the probability of x occurrences in an interval f(x) = the 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 x ! = x ( x – 1)( x – 2)... (2)(1) x ! = x ( x – 1)( x – 2)... (2)(1)

44. 44 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Poisson Probability Distribution n Poisson Probability Function In practical applications, x will eventually become In practical applications, x will eventually become large enough so that f ( x ) is approximately zero large enough so that f ( x ) is approximately zero and the probability of any larger values of x and the probability of any larger values of x becomes negligible. becomes negligible. In practical applications, x will eventually become In practical applications, x will eventually become large enough so that f ( x ) is approximately zero large enough so that f ( x ) is approximately zero and the probability of any larger values of x and the probability of any larger values of x becomes negligible. becomes negligible. Since there is no stated upper limit for the number Since there is no stated upper limit for the number of occurrences, the probability function f ( x ) is of occurrences, the probability function f ( x ) is applicable for values x = 0, 1, 2, … without limit. applicable for values x = 0, 1, 2, … without limit. Since there is no stated upper limit for the number Since there is no stated upper limit for the number of occurrences, the probability function f ( x ) is of occurrences, the probability function f ( x ) is applicable for values x = 0, 1, 2, … without limit. applicable for values x = 0, 1, 2, … without limit.

45. 45 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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?

46. 46 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Poisson Probability Distribution  = 6/hour = 3/half-hour, x = 4 n Example: Mercy Hospital Using the probabilityfunction probabilityfunction

47. 47 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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, 13 … Actually, the sequence continues: 11, 12, 13 … n Example: Mercy Hospital

48. 48 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

49. 49 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Poisson Probability Distribution Variance for Number of Arrivals During 30-Minute Periods  =  2 = 3 n Example: Mercy Hospital

50. 50 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

51. 51 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Hypergeometric Probability Function Hypergeometric Probability Distribution where: x = number of successes n = number of trials n = number of trials f ( x ) = probability of x successes in n trials f ( x ) = probability of x successes in n 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

52. 52 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 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 n – x failures can be selected from a total of N – r failures in the population number of ways n elements can be selected from a population of size N from a population of size N number of ways n elements can be selected from a population of size N from a population of size N

53. 53 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypergeometric Probability Distribution n Hypergeometric Probability Function If these two conditions do not hold for a value of If these two conditions do not hold for a value of x, the corresponding f ( x ) equals 0. x, the corresponding f ( x ) equals 0. If these two conditions do not hold for a value of If these two conditions do not hold for a value of x, the corresponding f ( x ) equals 0. x, the corresponding f ( x ) equals 0. However, only values of x where: 1) x < r and However, only values of x where: 1) x < r and 2) n – x < N – r are valid. 2) n – x < N – r are valid. However, only values of x where: 1) x < r and However, only values of x where: 1) x < r and 2) n – x < N – r are valid. 2) n – x < N – r are valid. The probability function f ( x ) on the previous slide The probability function f ( x ) on the previous slide is usually applicable for values of x = 0, 1, 2, … n. is usually applicable for values of x = 0, 1, 2, … n. The probability function f ( x ) on the previous slide The probability function f ( x ) on the previous slide is usually applicable for values of x = 0, 1, 2, … n. is usually applicable for values of x = 0, 1, 2, … n.

54. 54 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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?

55. 55 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

56. 56 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypergeometric Probability Distribution n Mean n Variance

57. 57 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Hypergeometric Probability Distribution Mean Mean Variance Variance n Example: Neveready’s Batteries

58. 58 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. ( N – n )/( N – 1) approaches 1. If the population size is large, the term ( N – n )/( N – 1) approaches 1. ( N – n )/( N – 1) approaches 1. 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 ). 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 ). Note that these are the expressions for the expected value and variance of a binomial distribution. value and variance of a binomial distribution. Note that these are the expressions for the expected value and variance of a binomial distribution. value and variance of a binomial distribution. continued

59. 59 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 distribution with n trials and a probability of success p = ( r / N ). success 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 distribution with n trials and a probability of success p = ( r / N ). success p = ( r / N ).

60. 60 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 5