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# STATISTICAL METHOD STA 291

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This 52 page Class Notes was uploaded by Helga Torp Sr. on Friday October 23, 2015. The Class Notes belongs to STA 291 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/228280/sta-291-university-of-kentucky in Statistics at University of Kentucky.

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Date Created: 10/23/15

STA 291 Summer 2008 Lecture 5 Casinos quotThere is no such thing as luck It is all mathematicsquot Casinos make money on their games because of the mathematics behind the games With a few notable exceptions the house always wins in the long run because ofthe mathematical advantage the casino enjoys over the player Because of a famous mathematical result called the law of large numbers a casino is guaranteed to win in the long run In the gambling industry nothing plays a more important role than mathematics STA 291 Lecture 5 2 6 Probability Abstract but necessary because this is the mathematical theory underlying all statistical inference Fundamental concepts that are very important to understanding Sampling Distribution Confidence Interval and PValue Our goal for Chapter 6 is to learn the rules involved with assigning probabilities to events STA 291 Lecture 5 Probability Basic Terminology Experiment Any activity from which an outcome measurement or other such result is obtained Random or Chance Experiment An experiment with the property that the outcome cannot be predicted with certainty Outcome Any possible result of an experiment Sample Space The collection of all possible outcomes of an experiment Event A specific collection of outcomes Simple Event An event consisting of exactly one outcome STA 291 Lecture 5 Experiments Outcomes Sample Spaces and Events Examples 1 Flip a coin 2 Flip a coin 3 times 3 Roll a die 4 Draw a SRS of size 50 from a population 5 Time your commuting time in a weekday morning 6 A football game between two chosen teams 7 Give AIDS patient a treatment and record how long heshe lives STA 291 Lecture 5 5 There are often more than one way to assign probability to a sample space could be infinitely many ways Statistical techniques we will learn later help identify which one quotreflecting the truth 0 In Chap 67 we do not identify which probability is more quottruequot but just learn the consequences of a legitimate probability assignment STA 291 Lecture 5 6 legitimate probability PA has to be a number between 0 and 1 inclusive The probability ofthe sample space S must be 1 PS 1 When we able to list a simple events 0 summation over i forPOl must be 1 STA 291 Lecture 5 2 POl1 PA could be computed by sum of POz over those 0 inside A Complement LetA denote an event The complement of an event A All the outcomes in the sample space S that do not belong to the event A The complement of A is denoted by AC Law of Complements PAC1 PA S Example If the probability of getting a working computer is 07 What is the probability of getting a defective computer STA 291 Lecture 5 9 Union and Intersection LetA and B denote two events The union of two events All the outcomes in S that belong to at least one of A or B The union ofA and B is denoted by AU B The intersection of two events All the outcomes in Sthat belong to both A and B The intersection ofA and B is denoted by AmB STA 291 Lecture 5 Additive Law of Probability Let A and B be two events in a sample space S The probability of the union of A and B is PAU B PA PB PAm B 0 STA 291 Lecture 5 11 Using Additive Law of Probability Example At a large University all rstyear students must take chemistry and math Suppose 15 fail chemistry 12 fail math and 5 fail both Suppose a rstyear student is selected at random What is the probability that this student failed at least one of the courses 0 STA 291 Lecture 5 12 Disjoint Events LetA and B denote two events Disjoint mutually exclusive events A and B are said to be disjoint if there are no outcomes common to both A and B Using notation this is written as A m B Q Note The last symbol denotes the null set or the empty set STA 291 Lecture 5 13 Disjoint Events Let A and B be two events in a sample space S The probability of the union of two disjoint mutually exclusive events A and B 15 PAUB PAPB STA 291 Lecture 5 S 14 Assigning Probabilities to Events The probability of an event is a value between 0 and 1 In particular 0 implies that the event will not occur 1 implies that the event will occur How do we assign probabilities to events STA 291 Lecture 5 Assigning Probabilities to Events There are different approaches to assigning probabilities to events Objective equally likely outcomes classical approach relative frequency Subjective STA 291 Lecture 5 16 Equally Likely Approach Laplace The equally likely outcomes approach usually relies on symmetrygeometry to assign probabilities to events As such we do not need to conduct experiments to determine the probabilities Suppose that an experiment has only n outcomes The equally likely approach to probability assigns a probability of 1n to each ofthe outcomes Further if an eventA is made up of m outcomes then ProbA mn STA 291 Lecture 5 17 Equally Likely Approach Examples 1 Select a SRS of size 2 from a population 2 Roll a fair die The probability of getting quot5 is 16 This does not mean that whenever you roll the die 6 times you definitely get exactly one quot5 The probability of the event quot4 or above is STA 291 Lecture 5 Counting method At every step you always have k choices There are m steps Total number ofchoices k k k k k kto m power Example pick 3 lotto 10 to 3 power STA 291 Lecture 5 19 Counting method Ranking of n distinct objects You first decide who is 1 then who is 2 etc Total choices nn 1n 2n 33 2 1 n STA 291 Lecture 5 20 Counting method I Number of ways to select a group of m out of n candidates no order nn 1n 2n m1 mm 1m 23 2 1 STA 291 Lecture 5 21 Nl Relative Frequency Approach von Mises The relative frequency approach borrows from calculus concept of limit Here s the process Repeat an experiment n times Record the number of times an event A occurs Denote that value by a Calculate the value an STA 291 Lecture 5 22 Relative Frequency Approach We could then define the probability of an ProbltAgt lim event A in the following n oo n manner Typically we can t can t do the quotn to in nity in a reallife situations so PI ObA instead we use a quotlargequot 72 n and say that STA 291 Lecture 5 23 Example Relative Frequency Approach Relative frequency of an event occurring in an in nitely large number of trials Online Applet Coin Tossing STA 291 Lecture 5 Time Period Number of Male Total Number of Live Relative Frequency of Live Births Births Live Male Birth 1965 1927054 3760358 051247 19651969 9219202 17989360 051248 19651974 17857860 34832050 051268 24 Relative Frequency Approach What is the formal name of the device that allows us to use quotlargequot n Law of Large Numbers As the number of repetitions of a random experiment increases the chance that the relative frequency of occurrence for an event will differ from the true probability of the event by more than any small number approaches 0 STA 291 Lecture 5 25 Subjective Probability Approach A subjective probability relies on a person to make a judgment as to how likely an event will occur The events of interest are usually events that cannot be replicated easily or cannot be modeled with the equally likely outcomes approach As such these values will most likely vary from person to person The only rule for a subjective probability is that the probability of the event must be a value in the interval 01 STA 291 Lecture 5 26 Probabilities of Events LetA be the eventA 01 oz ok where 01 oz ok are k different outcomes Then PAP01P0239 P0k Problem The number on a license plate is any digit between 0 and 9 What is the probability that the rst digit is a 3 What is the probability that the rst digit is less than 4 STA 291 Lecture 5 27 Probability tables One row of outcomes one row of corresponding probabilities R x C probability tables when the outcomes are classified by two features STA 291 Lecture 5 28 Example Smoking and Lung Disease Lung Not Lung Marginal Disease Disease smoke status Smoker 012 019 Nonsmoker 003 066 Marginal disease status STA 291 Lecture 5 29 Equivalent to a table with 4 entries smoker amp lung disease 012 smoker amp not lung disease 019 nonsmoker amp lung disease 003 nonsmoker amp not lung disease 055 But the R x C table reads much better STA 291 Lecture 5 3O From the R x C table we can get a table for smoker status alone or disease status alone Those are called marginal probabilities STA 291 Lecture 5 31 One way street Given the joint probability table we can figure out the marginal probability Given the marginal we may not determine the joint there can be several different joint tables that lead to identical marginal STA 291 Lecture 5 32 Example Smoking and Lung Disease Lung Not Lung Marginal Disease Disease smoke status Smoker 002 029 Nonsmoker 013 056 Marginal disease status Same marginal differentjoint STA 291 Lecture 5 33 Using the table Psmoker and lung disease002 Psmoker or lung disease STA 291 Lecture 5 34 Conditional Probability PA m B PA B PB provided PB 72 0 Note PAB is read as quotthe probability that A occurs given that B has occurred Note PA and B PABPB STA 291 Lecture 5 35 Independence f events A and B are independent then the events A and B have no influence on each other So the probability ofA is unaffected by whether B has occurred Mathematically ifA is independent of B we write PAB PA STA 291 Lecture 5 36 Multiplication Rule and Independent Events Multiplication Rule for Independent Events LetAand B be two independent events then PAm B PAPB Examples Flip a coin twice What is the probability of observing two heads Flip a coin twice What is the probability of getting a head and then a tail A tail and then a head One head Three computers are ordered If the probability of getting a working computer is 07 what is the probability that all three are working STA 291 Lecture 5 37 Terminology PAn BPAB Joint probability of A and B of the intersection ofA and B PAB PA Conditional probability ofA given B Marginal probability of A STA 291 Lecture 5 38 Example Medical Screening Screenings are routinely performed on patients Examples PSA for men determining evidence of prostate cancer Pap for women cervical cancer These screening tests are not 100 accurate Falsepositive test result The patient does not have the disease but the test shows positive Falsenegative result The patient does have the disease but the test produces a negative result STA 291 Lecture 5 39 Example Medical Screening The Pap smear is the standard test for cervical cancer Falsepositive rate 0636 Falsenegative rate 0180 Family history and age are factors that must be considered when assigning a probability of cervical cancer Suppose that the proportion of women a patient s age and with her family history that have cervical cancer is 2 Determine the effects a positive and negative Pap smear test have on the probability that the patient has cervical cancer STA 291 Lecture 5 40 Example A1 A2 B1 20 15 Bz 60 05 Determine whetherthe events are independentfrom the following joint probabilities Example Cash Credit Card Debit Card Under 20 09 03 04 20100 05 21 18 Over 100 03 23 14 a What proportion of purchase was paid by debit card b Find the probability that a credit card purchase was over 100 c Determine the proportion of purchases made by credit card or by debit card 7 Random Variables A variable X is a random variable if the value thatX assumes at the conclusion of an experiment cannot be predicted with certainty in advance There are two types of random variables Discrete the random variable can only assume a finite or countably infinite number of different values Continuous the random variable can assume all the values in some interval STA 291 Lecture 5 43 Examples Which of the following random variables are discrete and which are continuous a X Number of houses sold by real estate developer per week b X Number of heads in ten tosses of a coin c X Weight of a child at birth d X Time required to run a marathon STA 291 Lecture 5 44 Properties of Discrete Probability Distributions De nition A Discrete probability distribution is just a list of the possible values of a rV X say Xi and the probability associated with each PXXi Properties lAll probabilities nonnegative 2Pr0babilities sum to OSPxlSl ZPxll STA 291 Lecture 5 45 Example The table below gives the of days of sick leave for 200 employees in a year Days 0 1 2 3 4 5 6 7 Number of 20 40 40 30 20 10 10 30 Employees An employee is to be selected at random and let X days of sick leave a Construct and graph the probability distribution ofX b Find PXs3 c Find PX23 d Find P3 s X s 6 STA 291 Lecture 5 46 Population Distribution vs Probability Distribution If you select a subject randomly from the population then the probability distribution for the value ofthe random variable X is the population distribution ofthat variable 0 Example Xnumber of sick daysheightgrade of a randomly chosen person STA 291 Lecture 5 47 Cumulative Distribution Function De nition The cumulative distribution function or CDF is Fx PXS x Motivation Some parts of the previous example would have been easier with this next tool Properties 1F0r any value x O S Fx 3 1 21fx1 lt x2 then Fx1 S Fx2 3Foo O and Foo 1 STA 291 Lecture 5 48 Example Let X have the following probability distribution x 2 4 6 8 10 Px 05 20 35 30 10 a Find PXs 6 b Graph the cumulative probability distribution function c Find PX gt 6 STA 291 Lecture 5 49 Expected Value of a Random Variable The Expected Value or mean of a random variable X is Mean EX u ZxPX 9 Example X 10 Px 05 20 35 30 10 What is E00 STA 291 Lecture 5 50 Expected Profit Example The L M Corporation purchases old rundown buildings remodels them and then sells them The corporation has the opportunity to purchase a building for 100000 which will cost 60000 to remodel The manager thinks there is a 50 chance that the building will sell for 120000 yielding a 40000 loss a 20 chance that the building will sell for 180000 yielding a 20000 profit and a 30 chance that the building will sell for 230000 yielding a 70000 profit What is the expected profit STA 291 Lecture 5 51 Variance of a Random Variable Variance VarX E 2 02 206 2PX xi Example x 2 4 6 8 10 Px 05 20 35 30 10 What is VarOO STA 291 Lecture 5 52

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