1 1 Slide © 2004 Thomson/South-Western Slides Prepared by JOHN S. LOUCKS St. Edward’s University Slides Prepared by JOHN S. LOUCKS St. Edward’s University

2 2 Slide © 2004 Thomson/South-Western Chapter 3, Part A Discrete Probability Distributions n Random Variables n Discrete Probability Distributions n Expected Value and Variance n Binomial Probability Distribution n Poisson Probability Distribution

3 3 Slide © 2004 Thomson/South-Western A random variable is a numerical description of the A random variable is a numerical description of the outcome of an experiment. outcome of an experiment. A random variable is a numerical description of the A random variable is a numerical description of the outcome of an experiment. outcome of an experiment. Random Variables A discrete random variable may assume either a A discrete random variable may assume either a finite number of values or an infinite sequence of finite number of values or an infinite sequence of values. values. A discrete random variable may assume either a A discrete random variable may assume either a finite number of values or an infinite sequence of finite number of values or an infinite sequence of values. values. A continuous random variable may assume any A continuous random variable may assume any numerical value in an interval or collection of numerical value in an interval or collection of intervals. intervals. A continuous random variable may assume any A continuous random variable may assume any numerical value in an interval or collection of numerical value in an interval or collection of intervals. intervals.

4 4 Slide © 2004 Thomson/South-Western Let x = number of TVs sold at the store in one day, Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) where x can take on 5 values (0, 1, 2, 3, 4) Let x = number of TVs sold at the store in one day, Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4) where x can take on 5 values (0, 1, 2, 3, 4) Example: JSL Appliances n Discrete random variable with a finite number of values

5 5 Slide © 2004 Thomson/South-Western Let x = number of customers arriving in one day, Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... where x can take on the values 0, 1, 2,... Let x = number of customers arriving in one day, Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2,... where x can take on the values 0, 1, 2,... Example: JSL Appliances n Discrete random variable with an infinite sequence of values We can count the customers arriving, but there is no We can count the customers arriving, but there is no finite upper limit on the number that might arrive.

6 6 Slide © 2004 Thomson/South-Western Random Variables 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) Discrete

7 7 Slide © 2004 Thomson/South-Western The probability distribution for a random variable The probability distribution for a random variable describes how probabilities are distributed over describes how probabilities are distributed over the values of the random variable. the values of the random variable. The probability distribution for a random variable The probability distribution for a random variable describes how probabilities are distributed over describes how probabilities are distributed over the values of the random variable. the values of the random variable. We can describe a discrete probability distribution We can describe a discrete probability distribution with a table, graph, or equation. with a table, graph, or equation. We can describe a discrete probability distribution We can describe a discrete probability distribution with a table, graph, or equation. with a table, graph, or equation. Discrete Probability Distributions

8 8 Slide © 2004 Thomson/South-Western The probability distribution is defined by a The probability distribution is defined by a probability function, denoted by f ( x ), which provides probability function, denoted by f ( x ), which provides the probability for each value of the random variable. the probability for each value of the random variable. The probability distribution is defined by a The probability distribution is defined by a probability function, denoted by f ( x ), which provides probability function, denoted by f ( x ), which provides the probability for each value of the random variable. the probability for each value of the random variable. The required conditions for a discrete probability The required conditions for a discrete probability function are: function are: The required conditions for a discrete probability The required conditions for a discrete probability function are: function are: Discrete Probability Distributions f ( x ) > 0  f ( x ) = 1

9 9 Slide © 2004 Thomson/South-Western n a tabular representation of the probability distribution for TV sales was developed. distribution for TV sales was developed. n Using past data on TV sales, … Example: JSL Appliances Number Number Units Sold of Days Units Sold of Days x f ( x ) x f ( x ) /200

10 Slide © 2004 Thomson/South-Western Example: JSL Appliances n Graphical Representation of the Probability Distribution Values of Random Variable x (TV sales) ProbabilityProbability

11 Slide © 2004 Thomson/South-Western Discrete Uniform Probability Distribution The discrete uniform probability distribution is the The discrete uniform probability distribution is the simplest example of a discrete probability simplest example of a discrete probability distribution given by a formula. distribution given by a formula. The discrete uniform probability distribution is the The discrete uniform probability distribution is the simplest example of a discrete probability simplest example of a discrete probability distribution given by a formula. distribution given by a formula. The discrete uniform probability function is The discrete uniform probability function is f ( x ) = 1/ n where: n = the number of values the random variable may assume variable may assume the values of the random variable random variable are equally likely are equally likely

12 Slide © 2004 Thomson/South-Western Expected Value and Variance The expected value, or mean, of a random variable The expected value, or mean, of a random variable is a measure of its central location. is a measure of its central location. The expected value, or mean, of a random variable The expected value, or mean, of a random variable is a measure of its central location. is a measure of its central location. The variance summarizes the variability in the The variance summarizes the variability in the values of a random variable. values of a random variable. The variance summarizes the variability in the The variance summarizes the variability in the values of a random variable. values of a random variable. The standard deviation, , is defined as the positive The standard deviation, , is defined as the positive square root of the variance. square root of the variance. The standard deviation, , is defined as the positive The standard deviation, , is defined as the positive square root of the variance. square root of the variance. Var( x ) =  2 =  ( x -  ) 2 f ( x ) E ( x ) =  =  xf ( x )

13 Slide © 2004 Thomson/South-Western n Expected Value of a Discrete Random Variable Example: JSL Appliances expected number of TVs sold in a day x f ( x ) xf ( x ) x f ( x ) xf ( x ) E ( x ) = 1.20 E ( x ) = 1.20

14 Slide © 2004 Thomson/South-Western n Variance and Standard Deviation of a Discrete Random Variable of a Discrete Random Variable Example: JSL Appliances x -  ( x -  ) 2 f(x)f(x)f(x)f(x) ( x -  ) 2 f ( x ) Variance of daily sales =  2 = x TVssquared Standard deviation of daily sales = TVs

15 Slide © 2004 Thomson/South-Western n Formula Worksheet Using Excel to Compute the Expected Value, Variance, and Standard Deviation

16 Slide © 2004 Thomson/South-Western n Value Worksheet Using Excel to Compute the Expected Value, Variance, and Standard Deviation

17 Slide © 2004 Thomson/South-Western 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 Binomial Probability Distribution

18 Slide © 2004 Thomson/South-Western 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 © 2004 Thomson/South-Western where: where: f ( x ) = the probability of x successes in n trials f ( x ) = the probability of x successes in n trials n = the number of trials n = the number of trials p = the probability of success on any one trial p = the probability of success on any one trial Binomial Probability Distribution n Binomial Probability Function

20 Slide © 2004 Thomson/South-Western 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 © 2004 Thomson/South-Western Example: Evans Electronics n Binomial Probability Distribution Evans is concerned about a low retention rate for employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year. Evans is concerned about a low retention rate for employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year.

22 Slide © 2004 Thomson/South-Western Example: Evans Electronics n Binomial Probability Distribution Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year? Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year? Let : p =.10, n = 3, x = 1

23 Slide © 2004 Thomson/South-Western n Tree Diagram Example: Evans Electronics 1 st Worker 2 nd Worker 3 rd Worker x x Prob. Leaves (.1) Leaves (.1) Stays (.9) Stays (.9) Leaves (.1) S (.9) Stays (.9) S (.9) L (.1)

24 Slide © 2004 Thomson/South-Western Using Excel to Compute Binomial Probabilities n Formula Worksheet

25 Slide © 2004 Thomson/South-Western n Value Worksheet Using Excel to Compute Binomial Probabilities

26 Slide © 2004 Thomson/South-Western Using Excel to Compute Cumulative Binomial Probabilities n Formula Worksheet

27 Slide © 2004 Thomson/South-Western n Value Worksheet Using Excel to Compute Cumulative Binomial Probabilities

28 Slide © 2004 Thomson/South-Western Binomial Probability Distribution n Expected Value n Variance n Standard Deviation E ( x ) =  = np Var( x ) =  2 = np (1  p )

29 Slide © 2004 Thomson/South-Western n Binomial Probability Distribution Expected Value Expected Value Variance Variance Standard Deviation Standard Deviation Example: Evans Electronics E ( x ) =  = 3(.1) =.3 employees out of 3 Var( x ) =  2 = 3(.1)(.9) =.27

30 Slide © 2004 Thomson/South-Western A Poisson distributed random variable is often A Poisson distributed random variable is often useful in estimating the number of occurrences useful in estimating the number of occurrences over a specified interval of time or space over a specified interval of time or space A Poisson distributed random variable is often A Poisson distributed random variable is often useful in estimating the number of occurrences useful in estimating the number of occurrences over a specified interval of time or space over a specified interval of time or space It is a discrete random variable that may assume It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2,... ). an infinite sequence of values (x = 0, 1, 2,... ). It is a discrete random variable that may assume It is a discrete random variable that may assume an infinite sequence of values (x = 0, 1, 2,... ). an infinite sequence of values (x = 0, 1, 2,... ). Poisson Probability Distribution

31 Slide © 2004 Thomson/South-Western Examples of a Poisson distributed random variable: Examples of a Poisson distributed random variable: the number of knotholes in 24 linear feet of the number of knotholes in 24 linear feet of pine board pine board the number of knotholes in 24 linear feet of the number of knotholes in 24 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 © 2004 Thomson/South-Western 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 © 2004 Thomson/South-Western 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 = e =

34 Slide © 2004 Thomson/South-Western Example: Mercy Hospital n Poisson Probability Function Patients arrive at the emergency room of Mercy Hospital at the average rate of 6 per hour on weekend evenings. What is the probability of 4 arrivals in 30 minutes on a weekend evening? MERCY

35 Slide © 2004 Thomson/South-Western Example: Mercy Hospital n Poisson Probability Function  = 6/hour = 3/half-hour, x = 4

36 Slide © 2004 Thomson/South-Western Using Excel to Compute Poisson Probabilities n Formula Worksheet … and so on

37 Slide © 2004 Thomson/South-Western n Value Worksheet Using Excel to Compute Poisson Probabilities … and so on

38 Slide © 2004 Thomson/South-Western n Poisson Distribution of Arrivals Example: Mercy Hospital Poisson Probabilities Number of Arrivals in 30 Minutes Probability actually, the sequence continues: 11, 12, …

39 Slide © 2004 Thomson/South-Western Using Excel to Compute Cumulative Poisson Probabilities n Formula Worksheet … and so on

40 Slide © 2004 Thomson/South-Western n Value Worksheet Using Excel to Compute Cumulative Poisson Probabilities … and so on

41 Slide © 2004 Thomson/South-Western End of Chapter 3, Part A