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# PROB & STAT THEORY I MGS 9920

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This 82 page Class Notes was uploaded by Ms. Aaliyah Crist on Monday September 21, 2015. The Class Notes belongs to MGS 9920 at Georgia State University taught by Staff in Fall. Since its upload, it has received 52 views. For similar materials see /class/209860/mgs-9920-georgia-state-university in Managerial studies at Georgia State University.

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Date Created: 09/21/15

Chapter 3 Descriptive Statistics Numerical Measures Slide 1 Learning objectives 1 Single variable Part I Basic 11 How to calculate and use the measures of location 12 How to calculate and use the measures of variability 2 Single variable Part II Application 1 Understand What the measures of location eg mean median mode tell us about distribution sha e Discuss its use in manipulating simulated experiments 22 How to detect outliers using zscore and empirical rule 23 How to use Box plot to explore data 24 How to calculate weighted mean 25 How to calculate mean and variance for grouped data 3 Two variables 31 How to calculate and use the measures of association Covariance Correlation coef cient Slide 2 LO 1 Numerical measures Part1 I Numerical measures l Measures of Location 39 Mean median mode percentiles quartiles l Measures of Variability 39 Range interquartile range variance standard deviation coefficient of variation Slide 3 Numerical Measures If the measures are computed for data from a sample they are called sample statistics If the measures are computed for data from a population they are called population parameters A sample statistic is referred to as the point estilnator of the corresponding population parameter Slide 4 Median Mean WE Percenule Quamle l The m of a data set is the average of all the data values I The sample mean fis the point estimator of the population mean LL Sum of the values of the n observations Sum of the values of the N observations Number of observations in the sample Number of observations in the population Slide 5 39 Median Medlan WE Pzrcenule Quamle l The median of a data set is the value in the middle when the data items are arranged in ascending order 39 For odd number of observations I the median is the middle value 39 For even number of observations I the median is the average of the middle two values I Whenever a data set has extreme values the median is the preferred measure of central location 39 Often used in annual income and property value data Slide 6 Mode DI The mode of a data set is the value that occurs with the greatest frequency DI The greatest frequency can occur at two or more different values Percenule Quamle D 0 If the data have exactly two modes the data are bimodal D 0 If the data have more than two modes the data are multimoda Slide 7 L0 11 Mean Example mg 7233 l Q4 1 84 Compute the mean median and mode of the following sample 53 55 70 58 64 57 53 69 57 68 53 gtMean 59727 gtMedian 57 gtM0de 53 I What is the median if 59 is added to the data gtMedian 575 57582 Slide 8 Percentiles I A percentile provides information about how the data are spread over the interval from the smallest value to the largest value 0 Admission test scores for colleges and universities are frequently reported in terms of percentiles The gth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least 100 p percent of the items take on this value or more Percenule Quamle Slide 9 Percentiles Arrange the data in ascending order Compute index i the position of the pth percentile Percenule Quamle V If i is not an integer round up The pth percentile is the value in the ith position V If i is an integer the pth percentile is the average of the values in positions 139 and i1 Slide 10 Quar tiles Frag Quamlg D I Quartiles are specific percentiles D I First Quartile 25th Percentile D I Second Quartile 50th Percentile Median D I Third Quartile 75th Percentile Slide 11 Example Percentiles and Quartiles m3 Pzrcenule Quamle I Q4 p 84 Find 25th and 75th percentiles from the sample below 53 55 70 58 64 57 53 69 57 68 53 gt 25th percentile First quartile 53 gt 75th percentile Third quartile 68 Slide 12 Measures of Variability I It is often desirable to consider measures of variability dispersion as well as measures of location 0 For example in choosing supplier A or supplier B We might consider not only the average delivery time for each but also the variability in delivery time for each Range lnterquartile Range Variance Standard Deviation Coefficient of Variation Slide 13 LOR 12 Ran e 33 g Inefficient f DI The range of a data set is the difference between the largest and smallest data values DI It is the silnplest measure of variability DI It is very sensitive to the smallest and largest data values I Range of the sample 53 55 70 58 64 57 53 69 57 68 53 70 5317 Slide 14 Interquartile Range IQR D I The interguartile range of a data set is the mat difference between the third quartile and the first quartile D I It is the range for the middle 50 of the data D I It overcomes the sensitivity to extreme data values I IQR of the sample 53 55 70 58 64 57 53 69 57 68 53 68 53 15 Slide 15 Variance 237533 The variance is a measure of variability that utilizes all the data The variance is the average of the sguared differences between each data value and the mean The variance is computed as follows l 2 OZ L 1 ALL JV for a for a sample population Slide 16 Standard Deviation The standard deviation of a data set is the positive square root of the variance V It is measured in the same units as the data making it more easily interpreted than the variance V V The standard deviation is computed as follows for a population for a sample Slide 17 Coef cient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean The coefficient of variation is computed as follows for a sample for a population Slide 18 Example Variance Standard Deviation Ri i mm And Coef c1ent of Variation jggggfgggzn Consider the same data set 53 55 70 58 64 57 53 69 57 68 53 D l Variance l 7 Luliff r171 D l Standard Deviation the standard deviation is about 11 of of the mean mg n2 m C 39V m D l Coefficient of Variation v WI 2le A 1109J39m x 5973 Slide 19 LO 2 Numerical measure Part II D l Measures of Distribution Shape l Detecting Outliers 39 zscore empirical rule D l Exploratory Data Analysis D l The Weighted Mean and Working with Grouped Data Slide 20 10 Distribution Shape l Symmetric not skewed 39 Skewness is zero 39 Mean and median are equal V Relative Frequency i9 O L0 2 Shape vzrscure Emplncal Rule Gmuped data Slide 21 Distribution Shape l Moderately Skewed Left 39 Skewness is negative 39 Mean will usually be less than the median V Relative Frequency i9 O L0 2 Shape vzrscure Emplncal Rule Gmuped data Slide 22 11 L0 2 Shap2 Dlstrlbutlon Shape Emf Gmuped data I Moderately Skewed Right 39 Skewness is positive 39 Mean will usually be more than the median V Relative Frequency i9 O Slide 23 Distribution Shape Gmuped data I Highly Skewed Right 39 Skewness is positive often above 10 39 Mean will usually be more than the median 35 Pub 010 t 1 V Relative Frequency H L O O 08 Slide 24 12 L0 2 Shape z Scores Explnratury Weightedmean Gmuped data D The z score is often called the standardized value D It denotes the number of standard deviations a data value X is from the mean Slide 25 L0 2 Shape z Scores Explnratury Weightedmean Gmuped data D I An observation s z score is a measure of the relative location of the observation in a data set D I A data value less than the sample mean will have a z score less than zero D I A data value greater than the sample mean will have a z score greater than zero D I A data value equal to the sample mean will have a z score of zero Slide 26 13 Empirical Rule For data having a bell shaped distribution L0 2 Shape vzrscure EmpIrIcal Rule Gmuped data D 6826 of the values of a normal random variable are within 1 standard deviation lof its mean D 9544 of the values of a normal random variable are within 2 standard deviations lof its mean D 9972 of the values of a normal random variable are within 3 standard deviations lof its mean Slide 27 Empirical Rule D 99 72 D n 95 44 I D r6826 I I I I I I I I I I I I D I I I I I I I 39u 39u 1 u 36 2 L0 2 Shape vzrscure EmpIrIcal Rule Gmuped data Slide 28 14 L0 2 Shape Detecting Outliers EmmeT ElZ D I An outlier is an unusually small or unusually large value in a data set D I A data value with a z score less than 3 or greater than 3 might be considered an outlier D I It might be 39 an incorrectly recorded data value 39 a data value that was incorrectly included in the data set 39 a correctly recorded data value that belongs in the data set Slide 29 L0 2 Sha 2 Exploratory Data Analy51s Empmggfg g Gmuped data I The techniques of exploratory data analysis consist of silnple arithmetic and easy to draw pictures that can be used to summarize data quickly 39 Five Number Summary 39 Box Plot Slide 30 15 Sample 53 55 70 58 64 57 53 69 57 68 53 Smallest Value 53 Five Number Summary as Third Quartile Largest Value 70 L0 2 Shape mama Empincal Rule Gmuped data Slide 31 l A box plot is based on a five number summary Lower limit 30 Box Plot Upper limit 05 L0 2 Shape mama Empincal Rule Gmuped data as 15IQR 15IQR L 202 llllll 836 44 52 60 68 76 at H 92 100 No whisker this side 2 smallest Value Q1 Q153 Q368 Q257 Largest Value 70 Slide 32 16 The Weighted Mean and 22 Emplncal Rule Working with Grouped Data 39BP M W Gmuped data I Weighted Mean I Mean for Grouped Data l Variance for Grouped Data l Standard Deviation for Grouped Data Slide 33 L0 2 Sha 2 Weighted Mean mphgfg g I When the mean is computed by giving each data value a weight that re ects its ilnportance it is referred to as a weighted mean I Class grade is usually computed by weighted mean weight When data values vary in ilnportance the analyst must choose the weight that best re ects the ilnportance of each value Slide 34 17 L0 2 Shape Welghted Mean Empmi f i Explnratury Welghtedmean Gmuped data D T LIIpr Ui where xi value of observation 139 wt weight for observation 139 Slide 35 gt DI Grouped Data mp2 EmmeT ElZ The weighted mean computation can be used to obtain approxilnations of the mean variance and standard deviation for the grouped data To compute the weighted mean we treat the midpoint of each class as though it were the mean of all items in the class We compute a weighted mean of the class midpoints using the class freguencies as weights Silnilarly in computing the variance and standard deviation the class frequencies are used as weights Slide 36 18 L0 2 Shape Mean for Grouped Data Emgfga lz Gmuped data D l Sample Data 7 firVII n D l Population Data 39 CM 1 L 1 where 5 frequency of Classi M midpoint of class 139 Slide 37 L 2 s a Sample Mean for Grouped Data Empmi f i wg i ii Gmuped data Given below is the previous sample of monthly rents for 70 efficiency apartments presented here as grouped data in the form of a frequency distribution E D 600619 Slide 38 19 L0 2 Shape Sample Mean for Grouped Data Empmi f i V V V n n 0 This approxilnation differs by 241 from the actual sample mean of 49080 35665 21180 540559 2 5495 a t o 0 90 o Slide 39 L0 2 Shape Variance for Grouped Data Empmi f i Gmuped data D I For sample data Slide 40 L02 Sample Variance for Grouped Data Gmuped data V V V V WI gt2 n 324717 324795 0 0 a 560579 a continued Slide 41 L g 2 a Sample Variance for Grouped Data Emf wg i ii Gmuped data D l Sample Variance 52 20823429 70 1 301789 l Sam le Standard Deviation P 31101739 This approxilnation differs by only 20 from the actual standard deviation of 5474 Slide 42 21 LO 3 Measures of Association Between Two Variables l Covariance l Correlation Coefficient Slide 43 Cuvanance Covariance mam D The covariance is a measure of the linear association between two variables D Positive values indicate a positive relationship D Negative values indicate a negative relationship Slide 44 22 Cuvanance Covariance mum D The correlation coefficient is computed as follows D Zf vif jg7 fOr samples l Z ci JXyi fb for 39 7 i populations Slide 45 Cuvanance Correlation Coefficient mam D The coefficient can take on values between 1 and 1 D Values near 1 indicate a strong negative linear relationship Values near 1 indicate a strong positive linear D relationship Slide 46 Correlation Coefficient Cuvanance Currelatmn The correlation coefficient is computed as follows for samples populations Slide 47 V V Correlation Coefficient Cuvanance Currelatmn Correlation is a measure of linear association and not necessarily causation Just because two variables are highly correlated it does not mean that one variable is the cause of the other Slide 48 24 In Class Exercise I Q45 1 112 I Q46 1 112 Slide 49 End of Chapter 3 Slide 50 25 Chapter 4 Introduction to Probability Slide 1 Learning objectives 1 Introduction to probability 11 Understand experiments outcomes sample space 12 How to assign probabilities to outcomes 13 Understand events and how to assign probabilities to them 2 Basic relationships of probability 21 Understand and know how to compute complementary event union of two events intersection of two events mutually exclusive events 3 Conditional probability 31 Understand independent events 32 Understand multiplication law 4 Bayes Theorem 41 Tabular approach Slide 2 L o ucn39on lnn39cd An Experlment and Its Sample Space Rtm Bayes Thecl em Experiment outcomes Or Sample points Toss a coin Sample Space Slide 3 L o ucn39on quotne A551gn1ng Probablhtles Rm Bayes Thecl em D Classical Method Assigning probabilities based on the assumption of egually likely outcomes D Relative Frequency Method Assigning probabilities based on experilnentation or historical data D Subjective Method Assigning probabilities based on judgment Slide 4 L o ucn39on lnn39cd Subjectlve Method Rtm Bayes Thecl em D I When economic conditions and a company s circumstances change rapidly it might be inappropriate to assign probabilities based solely on historical data D I We can use any data available as well as our experience and intuition but ultimately a probability value should express our degree of belief that the experimental outcome will occur D I The best probability estimates often are obtained by combining the estimates from the classical or relative frequency approach with the subjective estimate Slide 5 L o ucn39on 2mm Events and Thelr Probablhtles Rtm j Bayes Thecl em D An event is a collection of sample points D The probability of any event is equal to the sum of the probabilities of the sample points in the event If we can identify all the sample points of an D experiment and assign a probability to each we can compute the probability of an event Slide 6 L o clion Events and Their Probabilities ti i i j Bayes Thecl em KPampL project page 152 Event C the project is completed in 10 months or less 2 6 2 7 2 8 3 6 3 7 4 6 PC P2 6 P2 7 P2 8 P3 6 P3 7 P4 6 15 15 05 10 20 05 Slide 7 Lo IJmF39a luc on Some Bas1c Relationships of Probability Retmrj Bayes Thecl em There are some basic probability relationships that can be used to compute the probability of an event without knowledge of all the sample point probabilities D Complement of an Event I D Union of Two Events D Intersection of Two Events I D Mutually Exclusive Events I Slide 8 0L 0 Introduction oRelationship of prob oConditional prob Bayes Theorem Venn diagram Complement of A AC Union of two events A U B l Intersection of Two Events A H B l Mutually Exclusive Events A H B 2 null set 0L 0 Introduction oRelationship of prob oConditional prob Addition Law Bayes Theorem D The addition law provides a way to compute the probability of event A or B or both A and B occurring The law is written as D PA U B PA PB PA m B The addition law for mutually exclusive events is PA o B PA PB there s no need to include PA o B Slide 10 L o ucn39on lnn39cd Condltlonal Probablllty Rtm Bayes Thecl em The probability of an event given that another event D has occurred is called a conditional probability The conditional probability of A given B is denoted D by PAB A conditional probability is computed as follows Slide 11 L o m cduc on J Multlpllcatron Law Rim i Bayes Thecl em The multiplication law provides a way to compute the probability of the intersection of two events The law is written as PA n B PBPA B V Slide 12 L o clion lnn39cdu Independent Events Rtm Bayes Thecl em If the probability of event A is not changed by the existence of event B we would say that events A and B are independent V Two events A and B are independent if V PA B PA or PBAPB Slide 13 Lo Multiplication Law Rda a jfr Ccl39n oralpl ob for Independent Events WSW The multiplication law also can be used as a test to see if two events are independent D The law is written as D PA n B PAPB Slide 14 Q30 p 167 Q31 p 167 L 0 lm cduc on Rda cnship ctpmb Ccndin39oral prob Bayes Thecl39em In class exercise Slide 15 L 0 ucn39on lnn39od Bayes Theorem Rm Bayes Thecl em D I Often we begin probability analysis with initial or prior probabilities D I Then from a sample special report or a product test we obtain some additional information D I Given this information we calculate revised or posterior probabilities D I Baxes theorem provides the means for revising the prior probabilities Prior Application New Posterior Information i gzgi Probabilities D Probabilities Slide 16 L o ucn39on lnn39cd Bayes Theorem Rtmcg ff Bayes Thecl em I Example L S Clothiers D A proposed shopping center for downtown businesses like L S Clothiers If the shopping 5 39 center is built the owner of r L S Clothiers feels it would be best to relocate to the center D The shopping center cannot be built unless a zoning change is approved by the town council The planning board must first make a recommendation for will provide strong competition or against the zoning change to the council Slide 17 Lo lnn cn lucn39on Bayes Theorem Rm Bayes Thecl em Prior Probabilities Let A1 town council approves the zoning change A2 town council disapproves the change Using subjective judgment PA1 7 PA2 3 Slide 18 L o clion lnn39cdu Bayes Theorem 43mm Bayes Thecl em New Information The planning board has recommended against the zoning change Let B denote the event of a negative recommendation by the planning board Given that B has occurred should L S Clothiers revise the probabilities that the town council will approve or disapprove the zoning change Slide 19 Bayes Theorem Bayes Thecl em Conditional Probabilities Past history with the planning board and the town council indicates the following PB A1 2 PB A2 9 D Hence PBC A1 8 PBC A2 1 Slide 20 10 L0 lnn39oduc on Bayes Theorem R mc EL E quot b sayes p prob Thecl39em Tree Diagram V v Town Council Planning Board Experimental Outcomes PA1nB 14 lt1 PBCIA18 PA1nB 56 lt1 PW B 27 lt39 I I I I I I I I I I I PAZnB 03 lt1 PBC A2 1 Slide 21 gt L o lm cduc on Rda cnship cl Ccndin39onal sayes Bayes Theorem prob prob Theme To find the posterior probability that event Al will occur given that event B has occurred we apply Bayes theorem PA1PBA1 2 PA1PB A1 PA2PB A2 PAnPB A Bayes theorem is applicable when the events for which we want to compute posterior probabilities are mutually exclusive and their union is the entire sample space Slide 22 11 L 0 clion lnn39cdu Bayes Theorem Rtm Bayes Thecl em Posterior Probabilities Given the planning board s recommendation not to approve the zoning change we revise the prior probabilities as follows D P IIB M A P B 1 A2 A2 D W Slide 23 I lm a lllclllioorA Bayes Theorem Rtm i Bayes Thecl em Conclusion The planning board s recommendation is good news for L S Clothiers The posterior probability of the town council approving the zoning change is 34 compared to a prior probability of 70 Slide 24 12 1 o lntl cn lucn39on Rda cnship ctpmb Tabular Approach w mm Bayes Thecl em Step 1 Prepare the following three columns Column 1 The mutually exclusive events for which posterior probabilities are desired Column 2 The prior probabilities for the events Column 3 The conditional probabilities of the new information given each event Slide 25 Tabular Approach Bayes Thecl em V V V 1 2 3 4 5 Prior Conditional Events Probabilities Probabilities A PltAigt PltBIAgt A1 7 2 A2 9 10 Slide 26 13 L0 lm cduc on Tabular Approach Rtm Bayes Thecl em Step 2 M Compute the joint probabilities for each event and the new information B by using the multiplication law Multiply the prior probabilities in column 2 by the corresponding conditional probabilities in column 3 That is PAZ B PAZ PB AZ Slide 27 Tabular Approach Bayes Thecl em V 1 2 3 4 5 Prior Conditional joint Events I quotquotquot I quotquotquot P quotquotquot Ai PAi PBlAi PAi B A1 7 2 14 A2 3 399 g 10 Slide 28 14 L 0 lnn39oduc on Tabular Approach t memb Cm1moml prob Bayes Theme Step 2 continued We see that there is a 14 probability of the town council approving the zoning change and a negative recommendation by the planning board There is a 27 probability of the town council disapproving the zoning change and a negative recommendation by the planning board Slide 29 Lo lm cduc on Tabular Approach Rtm i Bayes Thecl em Step 3 M Sum the joint probabilities The sum is the probability of the new information PB The sum 14 27 shows an overall probability of 41 of a negative recommendation by the planning board Slide 30 15 L o ucn39on Tabular Approach we 1 2 3 4 5 Prior Conditional Joint Events I quotquot39 I quotquot39 l quotquot39 Ai PAi PB IAi PAl B A1 7 2 14 A2 i 9 E 10 PB 41 Slide 31 Tabular Approach we Step 4 Column 5 Compute the posterior probabilities using the basic relationship of conditional probability PAl nB P PA lB The joint probabilities PAl H B are in column 4 and the probability PB is the sum of column 4 Slide 32 Tabular Approach Bayes Thecl em 1 2 3 4 5 Prior Conditional Joint Posterior Events I quot quot I quotquotquot quotquotquot I 39 Ai PAi PB IAi PAl n B PAi IB A1 7 2 14 3415 A2 3 9 6585 10 PB 41 Slide 33 In Class Exercise Bayes Thecl em Q59 p181 Q60 p181 Slide 34 17 End of Chapter 4 Slide 35 18 Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1 Understand random variables and probability distributions 11 Distinguish discrete and continuous random variables 2 Able to compute Expected value and Variance of discrete random variable 3 Understand 31 Discrete uniform distribution 32 Binomial distribution 33 Poisson distribution Slide 2 Lo Random variables ar Random Variables A random variable is a numerical description of the outcome of an experilnent V A discrete random variable may assume either a finite number of values or an infinite sequence of values V A continuous random variable may assume any numerical value in an interval or collection of intervals V Slide 3 Lo Random variables Example ISL Appliances l Discrete random variable with a finite number of values Let x number of TVs sold at the store in one day where x can take on 5 values 0 1 2 3 4 l Discrete random variable with an infinite sequence of values Let x number of customers arriving in one day where x can take on the values 0 1 2 We can count the customers arriving but there is no finite upper limit on the number that might arrive Slide 4 Lo Random variables ZEZandVamm Random Variables D Question Random Variable x Type D Family x Number of dependents Discrete size reported on tax return D Distance from x Distance in miles from Continuous home to store home to the store site Own dog x 1 if own no pet Discrete D or Cat 2 if own dogs only 3 if own cats only 4 if own dogs and cats Slide 5 Lo Random variables Pandmm Discrete Probability Distributions The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable V We can describe a discrete probability distribution with a table graph or equation V Slide 6 Lo Random variables 51333 d Discrete Probability Distributions l dist The probability distribution for discrete random D variable is defined by a probability function denoted by fx which provides the probability for each value of the random variable D The required conditions for a discrete probability function are x 2 0 W 1 Slide 7 md bl Ran omvzriz es 351313333323 m Dlscrete Probablllty Dlstrlbutlons dist D l Using past data on TV sales D l a tabular representation of the probability distribution for TV sales was developed Number Units Sold of Days E M 0 80 0 40 1 50 1 25 2 40 2 20 3 10 3 05 4 A 4 200 100 Slide 8 Lo Random variables WWW Discrete Probability Distributions l Graphical Representation of Probability Distribution Probability 0 o 2 3 4 Values of Random Variable at TV sales Slide 9 m Randum variables 331333 Expected Value and Varlance B lanlal dist Pmssun dist The expected value or mean of a random variable is a measure of its central location W u Ex x The variance summarizes the variability in the values of a random variable D Varx 02 2x u2fx The standard deviation 7 is defined as the positive square root of the variance V V Slide 10 to Randum variables ZEX D S Expected Value and Varlance B lanlal dist Pmssun dist l Expected Value a M M 0 40 00 1 25 25 D 2 20 40 3 05 15 4 10 expected number of TVs sold in a day Slide 11 to Randum variables zg g Expected Value and Varlance B lanlal dist Pmssun dist l Variance and Standard Deviation V V V V x x H x 02 x x MW 0 12 144 40 576 1 02 004 25 010 D 2 0 8 0 64 20 128 3 18 324 05 162 4 28 784 10 Variance of daily sales 72 1660 Standard deviation of daily sales 12884 TVs Slide 12 In class Exercise l Random variables 39 2 page 188 39 5 page 188 l Expected value and variance 39 16 page 196 39 17 page 197 Slide 13 m Randnmvanables Discrete Uniform Probability Distribution Pmssnndlst The discrete uniform probability distribution is the silnplest example of a discrete probability distribution given by a formula V The discrete uniform robabili function is the values of the random variable Where are equally likely 71 the number of values the random variable may assume Slide 14 Lo Randnm variables 15n j dm Binomial Distribution 39 39 St I Four Properties of a Binomial Experilnent 1 The experilnent consists of a sequence of 71 identical trials V D 2 Two outcomes success and failure are possible on each trial DJ The probability of a success denoted by p does D not change from trial to trial stationarity D 4 The trials are independent assumption Slide 15 Lo Randnm variables taxi fii idm Binomial Distribution 3921 d39 51 Our interest is in the number of successes occurring in the 71 trials We let x denote the number of successes occurring in the 71 trials Slide 16 Lo Randnm variables V 25m lI dst Binomial Distribution 39 3951 l Binomial Probability Function D r quot V IN nrz fw x P A P n z cl n x r0 A p 175 where fx the probability of x successes in 71 trials 71 the number of trials p the probability of success on any one trial Slide 17 Lo Randnm variables 25m lI dst Binomial Distribution 3921 d39 51 l Binomial Probability Function A A Probability of a particular sequence of trial outcomes with x successes in 71 trials Number of experilnental outcomes providing exactly x successes in 71 trials Slide 18 Lo Randnm variables Mama Example Evans Electronics lance Umfurm discrete dist B39Inom39lz ist Pmssun dist 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 estilnates a probability of 01 that the person will not be with the company next year V 0000000 Slide 19 EL l Using the Binomial Probability Function Choosing 3 hourly employees at random what is the probability that 1 of them will leave the company thisyear Let p10n3x1 j Ix 411751 5 A g D grim x Slide 20 0 Randumvanables x gj g t Example Evans Electromcs B39Inom39l2 ist PDIssDndIst 4 h l Tree D1agram v v V V 15 Worker 2 dWorker 3 dWorker g Prob I I I L 1 3 0010 Leaves 1 I I I 2 I o I I I 2 Stays 9 oquot I I lt I I I I I 2 Leaves 1 n I I Q I o I I Stays 9 o quot I I I S 0 7290 Slide 21 8574 1 41 7 160 1354 2430 3251 39 4320 0071 I 0574 0960 1406 39 7 2389 2880 0001 0010 0034 0080 0156 0270 0429 0640 1664 4084 3341 0911 Slide 22 11 0 Randumvanables 151313333 1 Blnomlal Dlstrlbution B39Inom39l2 ist Pmssun dist D l Expected Value Varx 72 np1 p D l Standard Deviation Ulnp1 p D l Variance Slide 23 D l Expected Value W U 31 employees out of 3 DI Variance Varx 02 319 D l Standard Deviation U 1rileee3 Slide 24 12 In class Exercise l 26 page 207 l 37 page 208 Slide 25 Lo Randnm variables andmm Poisson Distribution 1 A Poisson distributed random variable is often useful in estilnating the number of occurrences over a specified interval of tilne or space V It is a discrete random variable that may assume an infinite seguence of values X 0 1 2 V Slide 26 Lo Randnm variables Poisson Distribution Examples of a Poisson distributed random variable the number of knotholes in 14 linear feet of pine board the number of vehicles arriving at a toll booth in one hour Slide 27 L0 Randum variables mgrmvmmdm P01sson Distribution B lanlal dist l Two Properties of a Poisson Experilnent D l The probability of an occurrence is the same for any two intervals of equal length 2 The occurrence or nonoccurrence in any D interval is independent of the occurrence or nonoccurrence in any other interval Slide 28 14 m Randum variables EV andVanance 39 Umfmmdmg d P01sson Dlstrlbutlon B mumial ist Yo39lsson dist l Poisson Probability Function x L II a F r s x D 39x where fx probability of x occurrences in an interval u mean number of occurrences in an interval e 271828 Slide 29 L39O39d bl m mm as x g gm Example Mercy Hospltal 3233 D 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 Z l H mm E 1 Li I f g4 Slide 30 L0 Randnm variables V ngijg d Example Mercy Hospital dist l Using the Poisson Probability Function u 6 hour 3 half hour x 4 D Slide 31 Zl hnm variables MEN 351313333323 m Example Mercy Hospltal Il 11 7 a l Using Poisson Probability Tables u 23 21 39 28 1003 000 39 39 0672 0608 0550 16 quot Q 1815 1703 1596 2652 61 2l50 238l 39 2033 39 39739 2205 1169 39 39 39 39 39 1l88 39 39 0538 39 N 39 0801 0872 0910 39 0206 39 39 0362 0107 0155 0068 quot 0139 0163 0188 0015 0019 0031 0038 0017 0057 0068 Slide 32 L0 Randnm variables EV 1 V an anance Umfurm discrete dist B mnmia Yo39lsson dist l Poisson Distribution of Arrivals Example Mercy Hospital Poisson Probabilities 035 020 p E 015 actually 3 the sequence 010 continues 005 i i 11 12 000 l l l l l l l l l l l 0 1 2 3 4 5 6 7 8 9 10 Number of Arrivals in 30 Minutes Slide 33 L0 Randum variables 351313333323 m P01sson Dlstrlbutlon ii i iid t A property of the Poisson distribution is that D the mean and variance are equal Slide 34 17 Lo Randnmvanables mijgjgfggm Example Mercy Hospltal Bmumlaldls Yo39lsson dist l Variance for Number of Arrivals During 30 Mi11ute Periods Slide 35 In Class Exercise l 38 page 211 l 41 page 212 Slide 36 18 End of Chapter 5 Slide 37 19 Chapter 6 Continuous Probability Distributions Slide 1 Learning objectives 1 Understand continuous probability distributions 2 Understand Uniform distribution 3 Understand Normal distribution 31 Understand Standard normal distribution 32 Understand Normal approximation of binomial distribution 4 Understand Exponential distribution 41 Understand relationship between Poisson and Exponetial distribution Slide 2 Continuous Probability Distributions l A continuous random variable can assume any value in an interval on the real line or in a collection of intervals I Probability density function fx does not directly provide probability ie fC 0 for any c l Instead the area under the graph of fx corresponding to a given interval does provide the probability 39 ie we always compute the probability of interval for example between 10 and 20 Slide 3 Uniform Probability Distribution D l A random variable is uniformly distributed whenever the probability is proportional to the interval s length DI The uniform robabili densit function is D fx1b a foragxgb 0 elsewhere where a smallest value the variable can assume b largest value the variable can assume Slide 4 m Un39lform mammal Un1form Probablhty Dlstrlbutlon Nurmal Approximatan Expunenual DI Expected Value of x E06 a W2 Varx b a212 DI Variance of x Slide 5 m Uniform Smimml Un1form Probablhty Dlstrlbutlon NurmalAppruxlmatmn Expunenual l Example Slater39s Buffet D Slater customers are charged for the amount of salad they take Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces Slide 6 L0 Uniform Nurmal StandardNurmal Nurmal Apprtmmauun Expunenual l Uniform Probability Density Function Uniform Probability Distribution fx110 for55x515 D 0 elsewhere where x salad plate filling 39weight I Expected value and variance of X E06 a W2 Varx b a212 15 5212 833 5152 10 Slide 7 L0 Uniform Nurmal StandardNurmal Nurmal Apprtmmauun Expunenual Uniform Probability Distribution What is the probability that a customer will take between 12 and 15 ounces of salad f x V P12 5 x 5 15 1103 5 10 12 Salad Weight oz 15 Slide 8 20 Uniform 22221dwmmal In class exerc1se Nurmal Approximatan Expunenual I 1 p 228 I 2 p 228 Slide 9 20 Umfurm iiisii wml Normal Probablhty Dlstrlbutlon Nurmal Approximatan Expunenual l The normal probability distribution is the most iInportant distribution for describing a continuous random variable I It is widely used in statistical inference I It has been used in a wide variety of applications 39 Heights of people 39 Test scores 39 Amounts of rainfall Slide 10 to Umfurm 33le Normal Probablllty Dlstrlbutlon NurmalAppruximatmn Expunenual l Normal Probability Density Function 1 027 2 2 8706714 20 D fx where mean standard deviation 314159 271828 w lq Slide 11 to Umfurm liisii wml Normal Probablllty Dlstrlbutlon Nurmal Approximatmn Expunenual l Characteristics 39 The range of x is from 00 to 00 39 ymmetric skewness 0 39 The highest point Mean Median Mode 39 The total area under the curve is 1 5 to the left of the mean and 5 to the right x 00 Mean median mode Slide 12 to Umfurm liisii wml Normal Probablllty Dlstrlbutlon Nurmal Approximatan Expunenual l Characteristics continued 39 Defined by its mean u and its standard deviation 0 I Mean determines the center location of the curve I Standard deviation 7 determines how at the curve is larger values result in atter curves to Umfurm liisii wml Normal Probablllty Dlstrlbutlon Nurmal Approximatan Expunenual l Characteristics continued D 6826 of values of a normal random variable are within 1 standard deviation Iof its mean 9544 of values of a normal random variable are within 2 standard deviations of its mean 9972 of values of a normal random variable are within 3 standard deviations of its mean V V Slide 14 to Umfurm liisii wmml Normal Probablllty Dlstrlbutlon a Normal Approximation Expunenual l Characteristics continued D 9972 p r 9544 I D r 6826 I I I I I I I I I I I I I I I D I I I I I u Bc I u lc 39uu1c7 n36 u 26 u 26 Slide 15 L0 3533quot Standard Normal Z uiii mim Probability Distribution Expunenual A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a standard normal probabilitx distribution Converting Normal Distribution X to the Standard Normal Distribution Z xux 0 x Z Slide 16 2356mm Standard Normal utmal Z uiii mim Probability Distribution Expunenual l Example Pep Zone D Pep Zone sells auto parts and supplies including a popular multi grade motor oil When the stock of this oil drops to 20 gallons a replenishment order is placed Slide 17 Lo 3533quot Standard Normal Z uiii mim Probability Distribution Expunenual l Example Pep Zone D The store manager is concerned that sales are being lost due to stockouts while waiting for an order It has been determined that demand during 1 replenishment lead tilne is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons The manager would like to know the probability of a stockout Px gt 20 Slide 18 L0 Umfurm Nurmal Slzndzrd ormzl Nurmal Appraximatmn Expnnenual Standard Normal Probability Distribution N l Solving for the Stockout Probability D Step 1 Convert x to the standard normal distribution ZxH0 20 156 83 D Step 2 Find the area under the standard normal curve to the left of Z 83 see next sli e Slide 19 L0 Umfurm Nurmal Stzndzrd ormzl Nurmal Apprmumatmn Expunenua Standard Normal Probability Distribution N 1 l Cumulative Probability Table for the Standard Normal Distribution 07 08 6950 7051 7088 7123 7157 quot291 39 7389 7122 7151 7186 7011 39 39 7701 7731 7761 7791 7910 quot 8023 8051 8078 8186 8212 8238 39 8289 8315 8310 Slide 20 10 L0 3533quot Standard Normal Z uiii mim Probability Distribution Expunenual l Solving for the Stockout Probability curve to the right of Z 83 D Step 3 Compute the area under the standard normal Pz gt 83 1 Pz 5 83 1 7967 Probability of a stockout Slide 21 Lo 3533quot Standard Normal Z uiii mim Probability Distribution Expunenual l Solving for the Stockout Probability Area 1 7967 Area 7967 Slide 22 11 Standard Normal Z uiii mim Probability Distribution Expunenual l Standard Normal Probability Distribution If the manager of Pep Zone wants the probability of a stockout to be no more than 05 what should the reorder point be Slide 23 Lo 3533quot Standard Normal Z uiii mim Probability Distribution 1 Expunenua l Solving for the Reorder Point Area 9500 Area 0500 Slide 24 Lo 3533quot Standard Normal 37 23n Probability Distribution Expnnenual l Solving for the Reorder Point D Step 1 Find the Z Value that cuts off an area of 05 in the right tail of the standard normal distribution 9418 9525 9616 9693 We lollt up the complemt of w the tail area 1 05 95 Slide 25 Lo 3533quot Standard Normal Z uiiiii ilmii m Probability Distribution Expunenual l Solving for the Reorder Point D Step 2 Convert Z05 to the corresponding value of x x H 27050 15 16456 2487 or A reorder point of 25 gallons will place the probability of a stockout during leadtirne at slightly less than 05 Slide 26 13 u39mr39mm 1 mma Slzndzrd NurmalA Expunenua Standard Normal 13323 Probability Distribution 1 l Solving for the Reorder Point By raising the reorder point from 20 gallons to 25 gallons on hand the probability of a stockout decreases from about 20 to 05 This is a significant decrease in the chance that Pep Zone will be out of stock and unable to meet a customer s desire to make a purchase Slide 27 u39mr39mm Nurmal StandardNur mal Emma of Blnomlal Probablhtles Normzl A Expunenu D Normal Approximation I When the number of trials 71 becomes large evaluating the binomial probability function by hand or with a calculator is difficult D I The normal probability distribution provides an easy to use approximation of binomial probabilities where n gt 20 y 3 5 and n1 p 3 5 Slide 28 14 153 Normal Approximation StandardNurmal quotquot A fp quotquotm quot of Binomial Probabilities Expunenua DI Set unp 0 MPU P D I Add and subtract 05 a continuity correction factor because a continuous distribution is being used to approxilnate a discrete distribution For example Px 10 is approxilnated by P95 E x E 105 Slide 29 Umfurm 2223 1 In class exerc1se rmz Normzl Approxirnzl39lon Expunenual l Standard normal distribution 39 11 p 240 39 14 p 241 l Normal approxilnation 39 26 p 245 39 27 p 245 Slide 30 15 m Umfurm gggggidemal Exponentlal Probablhty Dlstrlbutron NurmalAppruximatmn onenlial l The exponential probability distribution is useful in describing the time it takes to complete a task I The exponential random variables can be used to describe Time between Time required Distance between vehicle arrivals to complete major defects at a toll booth a questionnaire in a highway Slide 31 L0f Ummm gggggidemal Exponentlal Probablhty Dlstrlbutron mmalAppmximatmn N onenlial l Density Function where u mean 6 271828 l Cumulative Probabilities PM x0 1 1quot 0W where x0 some specific value of x Slide 32 16 Eg ir im Exponential Probability Distribution onenlial l Example Al s Full Service Pump 3 D The tilne between arrivals of cars at Al s full service gas pump follows E an exponential probability distribution with a mean tilne between arrivals of 3 minutes Al would like to know the probability that the tilne between two successive arrivals will be 2 minutes or less Slide 33 Umfurm 33de Exponential Probability Distribution Nurmal Approximatan onenlial f x Px 5 2 1 271828 23 1 5134 4866 l Fivbe lllllllllllx 1234567891 Time Between Successive Arrivals Slide 34 17 to Umfurm 3mgde Exponentlal Probablllty Dlstrlbution Normal Approximatan onenlial w D A property of the exponential distribution is that the mean u and standard deviation 7 are equal D Thus the standard deviation 7 and variance 72 for the ti1ne between arrivals at Al s full service pump are Slide 35 L0 Umfurm le Relationship between the Poisson Standardemal E E ma mquot and Exponentlal Dlstrlbutlons The Poisson distribution provides an appropriate description of the number of occurrences per interval The exponential distribution provides an appropriate description of the length of the interval between occurrences Slide 36 18 to 152 Relationship between the Poisson Standardemal mm and Exponential Distributions Nurmal App Exponenlial l Poisson distribution 10X 710 fx x The average number of Cars that arrive at a Car wash during 1 hour 10 l The average tilne between cars arriving is 1 hour 10 cars 2 01 hourscar l The corresponding exponential distribution for the tilne between the arrivals fx ileixJ 210810x Slide 37 to Umfurm 2232mm In Class exerC1se NurmalAppruxlmatmn onenlial I 32 p 249 I 36 p 249 Slide 38 19 End of Chapter 6 Slide 39 20

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