PDF-1.2p=2Withxi+1= 1xi+ 2xi 1+i+1;startbyformingtheover-determinedsystem
Author : myesha-ticknor | Published Date : 2015-12-11
z b0BBBBBBx2x1x3x2xN1xN21CCCCCCA z A0121A z Unlikethepreviousp1casetryingtoexpressthesolutionATA1ATbanalyticallyisnottrivialWestartwithATA1266640x2x3xN1x1x2
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1.2p=2Withxi+1= 1xi+ 2xi 1+i+1;startbyformingtheover-determinedsystem: Transcript
z b0BBBBBBx2x1x3x2xN1xN21CCCCCCA z A0121A z Unlikethepreviousp1casetryingtoexpressthesolutionATA1ATbanalyticallyisnottrivialWestartwithATA1266640x2x3xN1x1x2. tXi=0g2iandkg(x)k1=maxi=0;:::;tjgij:Forapositivevaluer,wede netwocorrespondingtypesof\ball"centeredattheorigin:B2;N(r)=(N 1Xi=0aixi:N 1Xi=0a2ir2);B1;N(r)=(N 1Xi=0aixi: rair):Wehavetheusualinclusion @2xiXjj@2ui @xi@xjjthenitsatisesRosen'sconditionsinTheorem3.1.Denition5.Asociallyconcavegameisstrictsociallyconcaveifanyoneofthefollowingissatised:1.Foralliui(xi;x i)isstrictlyconcaveinxi.2.Theree case:E[s2]=E[1 NNXi=1(xi x)2]=1 x+NXi=1 x2]WeknowPNi=1xi=N xandPNi=1 x2=N x2.Plugtheseintothederivation:E[s2]=1 NE[NXi=1x2i 2N x2+N x2]=1 NE[NXi=1x2i N x2]=1 NE[NXi=1x2i] E[ x2]=E[x2] E[ x2]Acco Afteralittlealgebra,weobtainP3=1 1080@3922474816444524391A:Thensince =(1=3;1=3;1=3),wegetP(X3=j)=1Xi=0P(3)ij i=1 32Xi=0P(3)ij;forj=0;1;2:Inparticular,we ndthatP(X3=0)=132=324,P(X3=1)=62=324,andP(X3=2) @ 0= 2Pni=11Yi ( 0+ 1xi)]@S @ 1= 2Pni=11Yi ( 0+ 1xi)]xi(2.3)Whencomparedtozeroweobtainsocallednormalequations:8:Pni=1(b 0+b 1xi)=Pni=1YiPni=1(b 0+b 1xi)xi=Pni=1xiYi(2.4)Thissetofequationscanbewrittena Goldsman|ISyE673912.1SimpleLinearRegressionModelSupposewehaveadatasetwiththefollowingpairedobservations:(x1;y1);(x2;y2);:::;(xn;yn)Example:xi=heightofpersoniyi=weightofpersoniCanwemakeamodelexpressing E Thevalueof!0and!1minimizingQcanbederivedbydi!erentiatingQwithrespectto (Xi" y�b1 x(Note: y=1 nPni=1yiand x=1 nPni=1xi.)Proof:Ignoringthesecondnormalequation,startbydividingtherstnormalequationbyn:1 nnXi=1yi=b0+b1 nnXi=1xi:Rearrangingthisequation,andnotingthat y=1 nPni (8)Pni=1(xi� x)(yi� x y: Proof:Expand.e.g.,Pni=1xi(yi� y)intherighthandsideof(7),use(2)andthedenitionof x.(9)Pni=1(xi� x)2=Pni=1x2i�n x2: Proof:By(5)wegetthatPni=1(xi� x)2=P variancebutstillveryslightlybiasedintermsofstandarddeviation,writtenasfollows:~[X]=vuut 1 Ns�1NsXi=1(xi�~ps)2(8)Thismethodofestimatingprobabilisticquantitiesbystatisticsoveralargesampleofda EThevalueof0and1minimizingQcanbederivedbydierentiatingQwithrespecttoXiX2b01niYiiXib1Yb1Xiib1andb0areunbiasedestimatorsfor0Eb1iciEYiici01Xi0ici1iciXi00111Thereforeb1isanunbiasedXiei0andib0b1XiYib0ni1Xi Overview12ClassicalrigorousworkIThesecondmomentmethodIQuietplanting3Aphysics-inspiredrigorousapproachITheKauzmanntransitionIThefreeentropyinthe1RSBphase4Randomk-SATIArigorousBeliefPropagation-basedapp 16231618162416221627161716261625162016191614161616211615CorrespondingauthorJingTangThisworkislicensedundertheCreativeCommonsAttributionNonCommercialNoDerivatives40InternationalLicenseToviewacopyofthis 2fermions.Associatedwiththeelectronsisaconservedquantity,expressedasthequantumnumberknownastheleptonnumber.Theleptonnumberofthenegatronis,byconvention+1.Theleptonnumberofthepositron,alsotheanti-partic
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