Upload
others
View
12
Download
0
Embed Size (px)
Citation preview
PCB PCAABH tPCAnnB
PI cnn.IT C kIn anelements 2 subset
cxtyt cxtyl.ityI Eo 7 xiyn i
fI a kiIPCAIBtkftfp.BY PCANBt
PCAIBIPCBHKBIAIPCAlypa.yyy.PHPCH
PCBlAPCAlPCBIAHAHPCBIADPCADPCAIBtpPCBHKBHHPCAHPCBHHPCA.cl
If independentPCAAB PCA PCB PattB PCA PCBIAFPCB
If independent withHEB thenpathB PCA PCB but AEB thenpathB KAI
pattPCA PCBPCAFO or PCB PGH
Sequentialprobabilityproblems
KANHAMAN PCHHPCAIADXPCAHA.mxConditionalindependentif PLANBc p page butmutuallyindependence
conditionallyindependent
AxIBiAhBindicator CDFFix PDistribution pcaaxsbtPCXD PCH.atFabExes
1 discreteBernoulliDistribution Yo buildingblocks XnBernoulliCDBinomialDistribution RVXbe ofsuccess inn trials MxMqm XnBinomialEnpGeometricDistribution RVXbe offailuresuntil1stsuccess Pq X GeometricNegative
BinomialDistribution RVXbe oftrialsuntilrthsuccess ftp.flprqx rx megbinom
PossionDistribution RVXbe ofsuccessin continteralCtimeetc XuPoissonCx2 Continuous UsingCDFipcakeb FCb Fx a orPDTipcacxebfbatxu.MX4Uniform
uniformDistribution but VKasbEU XnUniformdUExponentialDistribution XnExponentialA
Gamma DistributionGumofExponentials iX GammacapCenter
pwidth
NormalDistribution Rvxerdistrabuamarruditscenterwithsomespread X Norma1dm6
wehaveCDFFxthenPDtfxcxtadF.ee
Gamma Distribution
For a l Tak CaDraDwehavePDFifx cuisine T a caDdenatFx Heldt Normal Distribution
MianterydistributionGinidthofdistribution
Expectation
E doesn'thavetobeoneofpossibleXvalue
ECIAFPCA
fECHFaxxPx dis
ECDFIxfxcxidxcconD.hu_atbXthenECatbxkatbECX7
VCXTECCXECXD7 ECXYM.ECD ECx5Variancetypicallydenotedby a
changeofvariable
methodsGeneralmethod
CDFmethod
PDFmethodContRvsinverseimage
ifpcxcBomdyhcxl.hnKYEA7 PCxEFEAD
if his FycytpckytPCYELo.yH pcxeLo.htysHPCXkhtytF.ch4yd
ifhis FycgtPCYeytPCYEEe.gl PCxeLhtyi.xII PCX7htyD tPCxEhtyD4 Talk'yD
tygtchEy1h'ChkyDI
2019 Midterm
ECI PCID
FkettleEat I s 4since t42,6 EtonSECO21.4
Fs PSes_PC4 PGz4 l Pts 7 1Fye 4 l eC DeHs11
b'aWtsiFCD FCa Eal Fca
FHHkxebtpcxeatblxsatPHYIY.fi Pf.aYajatbI Hothead
bWtsiPxCktDpI FCKD
fromportawehave Fcb f bCIHai FlathFla
are FlatbFca f bCtFlat
PxCb
bn B h Ffkt F FfH I 1
Back FCK FCKD FCDCIFCKil FCKD
of
I
JointPMTipx.yct.ytPCAxifytPLHxhfyy.tlMarginalPAAFi EyPC x yy fixone sumup
others
JointCDFi Fxyay PCXexyey
III IiIEIfI9IaigIx.c.otiniIss a csio it.say
harginalPDFif ftp.ycxgidyfycytfffx.ycxyidx
independentRVs PCXEA.YEBtPCXEDPCYEBCDTifx.ycx.glFxFop
iid iindependent identicallydistributed
Uscannotbeindependentiftheirjointareisnot rectangular I AxdetermineifXthewecancheckiffx.x.CHH fx.CxDfxz44
marginal
conditionaldistribution
KCX.YKAKX.yjepfPCQHEAncx.ge
Pax4EB
conditional PMF P CNDPcxxlyyt.MYEgY PMpYYyf
conditionalCDFifxiylxlytPCK.MYyconditionalPDF if yay k4D JSI
4 chargeofRVs is similar to ID Ex fx.ycx.gl444145 OY
PDFmethod Ext'tw Y whatisfz.wcz.ws
fz.wcz.us txikhIw weknowthatfz.wG.wt tfffhffz.mugand
IJChkz.MY
SUMYRVs X1kt hCZ.wtchicxy.hzcx.gl Cxty3xy7
orderstatisticsgRVs largestvalleyso JG.TKIzhyIdfIjIff Igf zyzyayxi.kz
ConvolutionMethod z y mdfEIIIIYIIEIzamd2IEIfy hez.meE.EEIfzcztf.fx.ycx.ENdxwI8xiiEFDdtgz.wcz.wy.fxychtz.us KEEFFE12
samefardiscrete TAFTorRVsX tntheKthorderstatisticXatheKthsmallestvariable and oexei.oc.ge 0k 14 OE Hµ_mindXnn_MaxHitn
or iidRushAnthedistributionofMaxXen isFcnkxtECDIndfcmcxtadzftcxh.hn I
thedistributionofminXen is
FHAt EIFHIndfcmcxtnu FGD.my
jxIY then ELgcxthCYD ECgu.DEChaprthmanentyxis Earltbhcentralmomentis ECµTcovarianceofXandYa's covenc ECXMACYMDCorrelation py
cae6 64
ifCora'D 0 7 444CovayKo qyy
it 0
CovCxic properties
CovCxix VCx
COCKYECxYMally
XIYecovcxiDobutcovcx.Y oHXIYVCX.tkVCXiHVCXzt2C0VCX.Xz
eg
MxcetECetxECM mE i.ewmttofind cxk.wehavemxc4thendiffktimesymxt.tl
mjkkeiomdlee t
oomdECEDf.IE f cxidxagzetxPxcx1MxLt
myCtk x y
conditionalExpectation
GivenRVYamdeventA.gl'Disafunction
ELGCYHHI2 ygodpcyhtlyqodindfcyllthrkylttlwhaef.CATFLYIN PC'kylYH
Discrete ftp.yeylContinuous ELGCYHAIffgodfghtldyZOELYHDffy.ggIda put
Aifysly
GivenRVYandx.gl'Disafunction
7iscreeeiEgc.DK 2ygoDpcyx1Continuous EEgc.DKffgodfglHdyandfyylxlH
tK
iIn tncxit.at
nlwhereEExn
MbutvCxnI 9bestondardnormaldistribution