Exam 1 – 115aBasic ProbabilityBasic Probability (cont.)Slide 4Summation NotationSummation PropertiesSlide 7Random VariablesRandom Variables (cont.)Expected ValueExpected Value (cont.)Conditional ProbabilityIndependent EventsIndependent Events (cont.)Bayes’ TheoremComputer ToolsComputer Tools (cont.)Slide 18Exam 1Formula SheetExam 1 – 115aBasic Probability•For any event E, •The union of two sets A and B, A B, includes items that are in either A or B.•The intersection, A B, is the set of all items in both A and B.•The union of two sets can be found by1)(0 EP)()()()( BAPBPAPBAP Basic Probability (cont.)•If E and F are mutually exclusive,–then –and because •The complement of E, EC, is the event that E does not happen–so –and , the sample space)()()( FPEPFEP 0)( FEP FE)(1)( EPEPCSEEC•DeMorgan’s LawsBasic Probability (cont.) )(1)()()(1)()(FEPFEPFEPFEPFEPFEPCCCCCC Correct Inco rrectE F = SE + F = SP(E) + P(F) = 0.8P(E) P(F) = 0.8P(EC) = 1 P(E) = 0.6 EC = 1 E = 0.6P(EC) = 0.6 P(E)C = 0.6Summation Notation•Summation notation is used to write series efficiently•The letter k is called the index of summation•The numbers 1 and n are the lower limit and upper limit of the summation, respectively1 2 31...nk nka a a a a Summation Properties1 1 1( )n n nk k k kk k ka b a b 1 1 1( )n n nk k k kk k ka b a b 1 1n nk kk kca c a Summation Properties1nkc cn1( 1)2nkn nk 21( 1) 2 16nkn n nk Random Variables•We may assign a number, X, to every event that arises from an experiment.•Random variables only take on numerical values!•Given an experiment, a random variable, X, and a number, x, the expression X=x stands for the event that an outcome occurs to which X is assigned the value x.•We may then speak of probabilities of these events, i.e. the probability that the event X=x is P(X=x).•The sum of the probabilities for all possible values of a random variable is equal to 1 (i.e. you have accounted for all possible outcomes of the sample space).Random Variables (cont.)Expected Value•The expected value of X, denoted E(X), is the sum of all values of X, weighted by their respective probabilities. That is:•If the experiment is repeated a large number of times, and the value of X is noted for each trial, then the average of these observed values will approach E(X).•Know the interpretations!!!!niiixXPxXE1)()(Expected Value (cont.)•It is usually convenient to set up a chart for all values of X. Each row corresponds to each value, the last column is the product of the previous 2 columns, and the expected value of X is the sum of this last column.x P(X=x) x.P(X=x)Conditional Probability•The conditional probability of an event E, given that an event F has happened, is denoted P(E|F).•If , then •Note, generally.0)( FP)()()|(FPFEPFEP)|()|( EFPFEP Independent Events•Sometimes knowing that F has happened does not change the probability of E, that is, . We then say that E and F are independent of each other.•Tests for independent events, E and F:)|()( FEPEP )()()()()|(FPEPFEPEPFEP•If E, F, and G are three events, and E and F are independent given that G has happened, then * this may be generalized to include more . than 2 independent eventsIndependent Events (cont.))|()|()|( GFPGEPGFEP Bayes’ Theorem•If the sets B1, B2, …, Bn partition the sample space, S, and A is any event, then•Also, for any k with , niiiBPBAPAP1)()|()(10 kniiikkkBPBAPBPBAPABP1)()|()()|()|(Computer Tools•Microsoft Word for word processing mathematics–know how to open the equation editor–know the equation editor window•Microsoft Excel–understand the layout of a spreadsheet (for example, you should know what cell I am referring to if I specify cell B11)–know how to insert a formula–know the database functions•DCOUNT•DAVERAGE•DMAXComputer Tools (cont.)•Microsoft Excel (cont.)–with each of those functions, you should recognize•what is a valid input for the function•the dialogue box that appears•how to indicate the input fields•what fields to leave blank (i.e. in DCOUNT)Computer Tools (cont.)Exam 1•Combination of multiple choice questions and mostly short answer problems•Scratch paper is available, so the only required items for you to bring are calculators and pencils•NO CELL PHONES!!!!! If any cell phones ring or buzz during the exam, you may lose credit on the exam.Formula Sheet FEPFPEPFEP )()( )(1 EPEPC CCCFEFE CCCFEFE xxXPxXEall)()(
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