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# STT 1810 STT 1810

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This 15 page Class Notes was uploaded by Cale Steuber on Friday October 2, 2015. The Class Notes belongs to STT 1810 at Appalachian State University taught by Ross Gosky in Fall. Since its upload, it has received 34 views. For similar materials see /class/217698/stt-1810-appalachian-state-university in Statistics at Appalachian State University.

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Date Created: 10/02/15

Chapter 7 Introduction to probability Statistics is sometimes de ned as the analysis of variation When variation occurs in the world then outcomes are uncertain For example if we have some money to invest we can invest it in the Stock Market At the end of 1 year how likely is it that our investment will have increased in value The outcomes seem to be our investment could increase decrease stay the same Probability gives us a framework to deal with these types of situations Our eventual goal is to use the rules of yrobabili to determine statistical signi cance That is using probability to help us draw conclusions about a large population from a sample What is probability Probability is often de ned as the longterm proportion of times an event would occur in an extremely large conceptually in nite number of observations A simple explanation of probability can be de nes as How often would this result occur if I repeated this process many times Consider the ipping of a coin Everyone intuitively realizes that the probability of getting heads on any coin ip is 05 However obviously any one coin ip is either a heads or a tails In addition to this many sequences of 10 coin ips would not contain exactly 5 heads and 5 tails But if we ipped the coin over and over many times we would expect our long term proportion ofheads to be 05 This long term proportion of occurrence is what we will think of as the probability of an event When we talk about probability we assume we are looking at a random event What is a random event It is simply an event that can take on one or more outcomes For example ipping a coin is a random event Sometimes it comes up heads sometimes tails What we want to do with probability is determine a what are the events which can occur eg heads or tails b how likely are the events for example heads and tails are equally likely so their probabilities are each 05 Def We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions STT1810 Course Notes ChaDters 7 and 8 Page 1 Probability Models For fun let s look at some basic examples of probability calculations You ll nd that you intuitively already understand most of the ideas we ll discuss in this chapter For examples start thinking of probabilities in the following situations What is o The probability of getting a king from a full deck of cards 0 The probability of rolling a 12 with two die 0 The probability of getting 3 consecutive heads in 3 coin ips o The probability of drawing 2 consecutive kings from a full deck of cards without replacing the rst card 0 The probability of having one girl and one boy if you have two children assume that a girl and boy are equally likely These are just exercises for fun But they get us thinking about probabilities The probability of an event answers the question How likely in the long run is the eventI observed from among all possible events that could have occurred STT1810 Course Notes Chapters 7 and 8 Page 2 Probability Models To write a probability model we need to establish 2 things 1 What are all the possible outcomes 2 What are the probabilities of all the outcomes mentioned above Basic Terms The sample space S of a random phenomenon is the set of all possible outcomes of the phenomenon EX Flip a fair coin 3 times 5 H1H2H3 T1T2T31H1T2T31T1H2T3 TITZHST1H2H31H1T2H31H1H2T339 An event is any outcome or set of outcomes of a random phenomenon So an event is a subset of the sample space EX an event in the above case could be Event 1 two heads occur in the three ips 9 T1H2H3 H1T2H3 H1H2T3 Event 2 tails occurs before the rst heads 9 T1T2T3 T1H2T3 T1T2H3 T1H2H3 We are most often interested in probabilities of events A probability model is a mathematical description of a random phenomenon consisting of 2 parts a sample space S and a method of assigning probabilities to events Rules for Valid Probability Models 1 For any event A its probability must be between 0 and 1 inclusive Symbolically For any event A 0 E PrA E 1 2 All possible outcomes together must have probability 1 Symbolically If S is the sample space all possible events then PrS 1 Example Consider rolling a die It seems rational to assign probability 16 to each possible outcome 1 through 6 So our probability model would be Value of Die 1 2 3 4 5 6 Probability 16 16 16 16 16 16 Note that the following would also be a valid probability model Value of Die 1 2 Probability 12 14 116 116 116 116 even though we don t perceive it to be realistic It is still a valid probability model because it upholds all the probability rules We may argue that it s the wrong probability model for this random phenomenon but we must conclude that it is still a valid probability model STT1810 Course Notes Chapters 7 and 8 Page 3 Rules that guide probability computations for events Cnmnlltino P 39 39 quotquot39 0ft 39 I A Events Def Two events whose outcomes don t affect each other are called independent events More speci cally events A and B are called independent if PrB PrB knowing that A happened Information about A s occurrence does not change the probability of B Example Consider ipping a coin two times De ne events A and B as follows Event A the rst coin ip is Heads Event B the 2quot01 coin ip is Heads Knowing if event A is true has no effect on the probability of event B Hence events A and B are said to be independent events Def Two events whose outcomes affect each other are called dependent events Example Drawing cards from a deck De ne events A and B as follows Event A lst card drawn is an ace Event B 2quot01 card drawn is an ace Assuming the card is not replaced after draw 1 PrB is affected by whether or not event A occurred If event A occurred PrB is 351 If event A did not occur PrB is 451 Events A and B are examples of dependent events Computing Probabilities ofIndependent Events When any number of events A B C N are independent probabilities of them occurring together can be computed by multiplying the probabilities of the individual events together PrA and B and C and N PrAxPrBxPrCx xPrN Example Earlier we saw the probability of 3 heads in a row was 18 earlier in the notes Let s verify this using our formula Let A heads on rst ip B heads on 2quot01 ip and C heads on 3ml ip Using our formula Prheads on all 3 ips PrA and B and C 12x12x12 18 STT1810 Course Notes Chapters 7 and 8 Page 4 Computing Probabilities ofDependent Events If two events A and B are dependent we can compute the probability of them occurring together as PrA and B PrA X PrB given that A has already occurred PrA X Pr B l A For example let A event that an ace is drawn on the rst draw B event that an ace is drawn on the second draw We know that PrA 452 Now once we know A has occurred there are 3 aces left in the deck and 51 total cards Hence PrB A 351 So PrA and B 452 X 351 12 2652 000452 Computing Probabilities ofDisjoint Events In general in this course we ll be considering simple events that don t occur together De nition Two events are said to be disjoint or mutually exclusive if they have no outcomes in common EXample Rolling a die Let event A die comes up an even number Let event B die comes up an odd number Events A and B are disjoint EXample Rolling a die Let event A die comes up 1 Let event B event the die comes up an odd number Events A and B are not disjoint because they can occur together If two events have no outcomes in common the probability that one or the other occurs is the sum of their individual probabilities Symbolically This means if A and B are disjoint aka mutually eXclusive events PrA or B PrA PrB Computational EXample Consider our card eXample from earlier Let s make it a bit more complicated Find the probability that the 1st card drawn is an Ace or a King Note that these are disjoint events So Pr1st is Ace or King Pr1st is Ace Pr1st is King 452 452 852 Seems simple enough STT1810 Course Notes Chapters 7 and 8 Page 5 When Events are Not Disjoint The more complicated rule PrA or B PrA PrB 7 PrA and B applies in this case but we will not focus on it in Stt18 10 The Complement Rule The probability that an event does not occur is 1 minus the probability that it does occur Symbolically If A is any event then PrA does not happen l PrA Example Letters are sent to 10 businesses Each letter is delivered independently with a 99 chance of success What is the chance that at least 1 of these letters does not reach its destination Solution Note that the complement of A at least 1 failure is Thus we can nd Prat least 1 failure l 7 Pr 1 7 STT1810 Course Notes Chapters 7 and 8 Page 6 Birthdays Examgle STT1810 Course Notes ChaDters 7 and 8 Page 7 Chapter 8 Random Variables A random variable is a numeric outcome of a random phenomenon Mathematically it is a mapping from a sample space S to a number X Let s see how this works Example 1 Flip a fair coin 3 times 5 H1H2H3 T1T2T31H1T2T31T1H2T3 TITZHST1H2H31H1T2H31H1H2T339 Two potential random variables might be X number of heads that occur X number of tails that occur prior to the rst head occurring There are 2 types of random variables 1 Discrete only integer outcomes are possible Examples number of siblings number of heads in 3 ips of a coin number of calls to a call center in 1 hour etc 2 Continuous fractional outcomes are possible More technically we say that any real number in some range is possible even if not exactly measurable Examples a person s true height inches time before bus arrives in seconds measured with a stopwatch Probability Models for Discrete Random Variables This is not focused on in SttlSlO But it is worth a quick mention Discrete random variables have a set of possible outcomes that can be easily summarized in a list The general approach to these models is to create such a list and to nd the probabilities of the items on that list Let s use the example where a coin is ipped 3 times and X the number of tails before the rst heads occurs S H1H2H3 T1T2T3H1T2T3T1H2T3 TszHsT1H2H3H1T2H3H1H2T3 Possible X Values Elements of S leading to this X Probability of X STT1810 Course Notes Chapters 7 and 8 Page 8 Probability Models for Continuous Random Variables Suppose the variable X could be measured to any degree of speci city desired Then any real number in a range of possibilities could occur For instance let X my height If measured at 69 in we know really that it is probably not exactly 690 inches Rather we are saying that X is between 6875 and 6925 inches Numeric random variables that can take any real number in a speci ed range are called continuous random variables Typically a speci c measurement of a continuous random variable implies that a value in some range has occurred for example the height measurement described above Most commonly density curves are used as probability models for continuous random variables De nition A density curve is a mathematical function of X that o is nonnegative for all X o the total area under the density curve is l which translates to 100 o for any two constants a and b Pra E X b area under density curve between a and b Interpreting a Density Curve Height of Curve at any X value 9 relative frequency of occurrences at that X Tall Density Curve 9 X values in that region are relatively common Short Density Curve 9 X values in that region are less common When the density curve is zero this suggests that X values in this region cannot occur Think of a density curve as a smooth curve through the tops of a histogram of a very very large population They may be skewed or symmetric just like histograms The following is an intuitive example of how these density curve works Ex Buses arrive at a bus stop every 10 minutes Supposing they are always on schedule ha let X time a person waits until a bus arrives at the stop assuming they show up randomly and don t know the schedule What would the density curve look like in this case Find the probability that the person has to wait longer than 7 minutes for the bus STT1810 Course Notes Chapters 7 and 8 Page 9 The density curve distribution in the previous problem has a name the Uniform distribution But the most common distribution we ll use is the Normal Distribution Terminology For any distribution we will denote its mean as u pronounced mu And its standard deviation will be denoted as o pronounced sigma The Normal Distribution The normal distribution is o A single peaked symmetric bell or mound shaped curve 0 Scaled by 2 parameters u and o where u denotes the mean of the distribution and 6 denotes the standard deviation 0 The most common distribution used in statistics for a continuous dataset Pictures of Normal Distributions Notation for the Normal Distributions We will denote a normal distribution as Nu 6 to indicate a speci c distribution For example a normal distribution with mean 1 and standard deviation 10 will be denoted as Nl 10 The Empirical Rule In a Nu 5 distribution approximately 0 68 of the observations fall Within u 1 16 within 1 standard deviation of the mean 0 95 of the observations fall Within u i 26 within 2 standard deviations of the mean 0 997 virtually all of the observations fall Within u i 36 within 3 standard deviations of the mean STT1810 Course Notes Chapters 7 and 8 Page 10 Example Lengths of pregnancies in days tend to follow a N266l6 distribution What is the probability a pregnancy will last longer than 282 days The empirical rule is useful in approximating proportions of observations in the normal distribution but it does not provide speci c answers to many questions For example if XN015 and you are asked to nd Pr15 lt X lt 5 you cannot do so using the empirical rule Hence you should not speci cally use this empirical rule often it s just a general rule of thum We will learn a more speci c method of computing proportions of observations falling in a given region Unlike the empirical rule this method works for all problems and in all situations STT1810 Course Notes Chapters 7 and 8 Page 11 The Standard Normal Distribution The standard normal distribution has mean u 0 and standard deviation 6 1 We will denote a variable which follows a N0 1 distribution as Z X Important fact If XNu c then Z T follows a N0l distribution This value Z is called a Zscore If X follows a Nu 5 distribution then we regularly convert from X to Z using the above formula so that we can easily determine proportions of observations falling into any region of the distribution Computing Standard Normal Proportions Proportions for a standard normal Z variable are listed in your textbook For values of Z between 7349 and 349 table Z gives proportions of observations to the left lower than that value of Z in a N0l distribution We can use this table in two ways 1 to nd proportions of observations falling in any speci ed region of the N0l distribution to nd percentile values for the N0l distribution Recall that the pth percentile of any distribution is the value of the distribution that contains p of the values below it For example the 603911 percentile of a distribution X is the value X such that 60 of the values ofX will be less then X Equot Let s see how to use the Standard Normal Table to answer these 2 types of questions STT1810 Course Notes Chapters 7 and 8 Page 12 Examgles of nding Progo ions 1 Equot W 5 U 9 STT1810 Course Notes ChaDters 7 and 8 Find PrZ lt 11 Find PrZ lt 11 Find PrZlt117 Find Pr z gt 2 Find Pr z gt 2 Find Pr 2 lt z lt1 Page 13 Examgles of Finding Percentiles 1 Find the 753911 percentile of the N0l distribution 2 Find the 1539h percentile of the N0l distribution 3 How large a value of Z is required to be in the top 20 of the N0l distribution 4 the value 2 such that 45 of the observations in a N0 1 distribution are less than 5 Find the value 2 such that 65 of the observations in a N0 1 distribution are greater than 2 STT1810 Course Notes ChaDters 7 and 8 Page 14 Using the Z Table to determine percentiles of any Normally Distributed Random Variable Often you may want to know a percentile value of a normally distributed random variable called X For example you may want to know what value of X is the 90 11 percentile value You can use the Z table to gure this out by 1 nd the corresponding percentile value for Z in the Z table Call this value 2 2 translate from Z to X using the formula X 52 p X is now the desired percentile for the distribution of X values Example Physics GPA s follow a N26 03 distribution how high does a student s GPA have to be to be in the top 10 ofhisher major STT1810 Course Notes Chapters 7 and 8 Page 15

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