第一讲 复习.ppt

计量经济学数理统计复习 1 数理统计复习 北京外国语大学国际商学院应惟伟 计量经济学数理统计复习 2 主要内容 期望Expectedvalue方差Variance协方差Covariance 计量经济学数理统计复习 3 DefinitionofE X theexpectedvalueofX EXPECTEDVALUEOFARANDOMVARIABLE Theexpectedvalueofarandomvariable alsoknownasitspopulationmean istheweightedaverageofitspossiblevalues theweightsbeingtheprobabilitiesattachedtothevalues 计量经济学数理统计复习 4 Theexpectedvalueturnsouttobe7 Actually thiswasobviousanyway Wesawintheprevioussequencethatthedistributionissymmetricalabout7 EXPECTEDVALUEOFARANDOMVARIABLE xipixipixipixipix1p1x1p121 362 36x2p2x2p232 366 36x3p3x3p343 3612 36x4p4x4p454 3620 36x5p5x5p565 3630 36x6p6x6p676 3642 36x7p7x7p785 3640 36x8p8x8p894 3636 36x9p9x9p9103 3630 36x10p10 x10p10112 3622 36x11p11x11p11121 3612 36Sxipi E X 252 36 7 计量经济学数理统计复习 5 AlternativenotationforE X E X mX Veryoftentheexpectedvalueofarandomvariableisrepresentedbym Ifthereismorethanonerandomvariable theirexpectedvaluesaredifferentiatedbyaddingsubscriptstom EXPECTEDVALUEOFARANDOMVARIABLE 计量经济学数理统计复习 6 Expectedvaluerules Therearethreerulesthatwearegoingtouseoverandoveragain Rule1E X Y Z E X E Y E Z Rule2E bX bE X wherebisaconstant Rule3E b bwherebisaconstant 计量经济学数理统计复习 7 DefinitionofE g X theexpectedvalueofafunctionofX Example Forexample theexpectedvalueofX2isfoundbycalculatingallitspossiblevalues multiplyingthembythecorrespondingprobabilities andsumming EXPECTEDVALUEOFAFUNCTIONOFARANDOMVARIABLE 计量经济学数理统计复习 8 PopulationvarianceofX TheexpectedvalueofthesquareddeviationisknownasthepopulationvarianceofX ItisameasureofthedispersionofthedistributionofXaboutitspopulationmean POPULATIONVARIANCEOFADISCRETERANDOMVARIABLE 计量经济学数理统计复习 9 E X2 m2 E X m 2 E X2 2mX m2 E X2 E 2mX E m2 E X2 2mE X m2 E X2 2m2 m2 E X2 m2 ALTERNATIVEEXPRESSIONFORPOPULATIONVARIANCE 计量经济学数理统计复习 10 PopulationvarianceofX Inequations thepopulationvarianceofXisusuallywrittensX2 POPULATIONVARIANCEOFADISCRETERANDOMVARIABLE 计量经济学数理统计复习 11 TworandomvariablesXandYaresaidtobeindependentifandonlyifE f X g Y E f X E g Y foranyfunctionsf X andg Y Specialcase ifXandYareindependent E XY E X E Y INDEPENDENCEOFTWORANDOMVARIABLES Asaspecialcase theexpectedvalueofXYisequaltotheexpectedvalueofXmultipliedbytheexpectedvalueofYifandonlyifXandYareindependent 计量经济学数理统计复习 12 Variancerules Rule1IfY V W Var Y Var V Var W 2Cov V W Rule2IfY bZ wherebisaconstant Var Y b2Var Z Rule3IfY b wherebisaconstant Var Y 0Rule4IfY V b wherebisaconstant Var Y Var V Wecangetproofoftheserulesbydefinitionofvariance however wehavealternativeproofinthelaterpart 计量经济学数理统计复习 13 Definitionofsamplecovariance SAMPLECOVARIANCE EXAMPLECALCULATION GivenasampleofnobservationsontwovariablesXandY thesamplecovarianceistheaverageoftheproductsoftheirdeviationsabouttheirsamplemeans 计量经济学数理统计复习 14 ObservationSY11517 2421615 003814 91464 5051518 006126 2971219 2381818 699127 21102042 06 19127 5020148 00 SAMPLECOVARIANCE EXAMPLECALCULATION ThetableaboveshowsS theyearsofschooling highestgradecompleted andY thehourlyearnings 计量经济学数理统计复习 15 SAMPLECOVARIANCE EXAMPLECALCULATION WewillcalculatethesamplecovarianceofSandY Herethedataareplottedinascatterdiagram 计量经济学数理统计复习 16 ObservationSYS Y S Y 11517 241 753 0165 27721615 002 750 7762 1333814 91 5 250 686 3 599464 50 7 25 9 72570 50351518 001 753 7766 6076126 29 1 25 7 9359 91871219 23 1 255 006 6 25781818 694 754 46621 2119127 21 1 25 7 0158 768102042 066 7527 836187 890 19127 50 1 25 6 7258 40620148 000 75 6 225 4 668Total265284 49305 888Average13 2514 22515 294 SAMPLECOVARIANCE EXAMPLECALCULATION Theproductsaresummedanddividedby20 thenumberofobservations Thesamplecovarianceisthus15 29 计量经济学数理统计复习 17 1 IfY V W Cov X Y Cov X V Cov X W 2 IfY bZ wherebisaconstant Cov X Y Cov X bZ bCov X Z Example Cov X 3Z 3Cov X Z 3 IfY b wherebisaconstant Cov X Y Cov X b 0Example Cov X 10 0 COVARIANCERULES 计量经济学数理统计复习 18 Example supposeY b1 b2ZCov X Y Cov X b1 b2Z Cov X b1 Cov X b2Z 0 Cov X b2Z b2Cov X Z COVARIANCERULES 计量经济学数理统计复习 19 1 IfY V W Cov X Y Cov X V Cov X W ProofofCOVARIANCERULES 计量经济学数理统计复习 20 ProofofCOVARIANCERULES 2 IfY bZ wherebisaconstant Cov X Y Cov X bZ bCov X Z 计量经济学数理统计复习 21 3 IfY b wherebisaconstant Cov X Y Cov X b 0 ProofofCOVARIANCERULES 计量经济学数理统计复习 22 ProofofVARIANCERULES VarianceRule1 IfY V W Var Y Var V Var W 2Cov V W Proof Var Y Cov Y Y Cov Y V W Cov Y V Cov Y W Cov V W V Cov V W W Cov V V Cov W V Cov V W Cov W W Var V Var W 2Cov V W NotethattheorderofthevariablesmakesnodifferencewhendefiningcovarianceandhenceCov W V isthesameasCov V W 计量经济学数理统计复习 23 ProofofVARIANCERULES VarianceRule2 IfY bZ wherebisaconstant Var Y b2Var Z Proof Var Y Cov Y Y Cov Y bZ bCov Y Z bCov bZ Z b2Cov Z Z b2Var Z 计量经济学数理统计复习 24 VarianceRule3 IfY b wherebisaconstant Var Y 0Proof Var Y Cov Y Y Cov b b 0 ProofofVARIANCERULES 计量经济学数理统计复习 25 ProofofVARIANCERULES VarianceRule4 IfY V b wherebisaconstant Var Y Var V Proof Var Y Var V b Var V Var b 2Cov V b Var V 计量经济学数理统计复习 26 ALTERNATIVEEXPRESSIONFORSAMPLECOVARIANCE Wenowsumvertically Thefirstsummationisthesummationofthefirsttermsinthenlines 计量经济学数理统计复习 27 ALTERNATIVEEXPRESSIONFORSAMPLECOVARIANCE Twooft