8
PCB PCAABH tPCAnnB PI cnn.IT C kIn a n elements 2 subset cxtyt cxtyl.it yI Eo 7 xiyn i f I a kiI PCAIBt kftfp.BY PCANBt PCAIBIPCBHKBIAIPCAlypa.yyy.PH PCH PCBlA PCAl PCBIAHAHPCBIADPCADPCAIBt pPCBHKBHHPCAHPCBHHPCA.cl If independent PCAA B P CA PCB Patt B P CA PCBIAFPCB If independent with HEB then path B P CA PCB but AEB then path B KAI patt PCA PCB PCAFO or PCB PGH

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Page 1: PCBIAHAHPCBIADPCADPCAIBt - GitHub Pages

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

Page 2: PCBIAHAHPCBIADPCADPCAIBt - GitHub Pages

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

Page 3: PCBIAHAHPCBIADPCADPCAIBt - GitHub Pages

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

Page 4: PCBIAHAHPCBIADPCADPCAIBt - GitHub Pages

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

Page 5: PCBIAHAHPCBIADPCADPCAIBt - GitHub Pages

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

Page 6: PCBIAHAHPCBIADPCADPCAIBt - GitHub Pages

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

Page 7: PCBIAHAHPCBIADPCADPCAIBt - GitHub Pages

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

Page 8: PCBIAHAHPCBIADPCADPCAIBt - GitHub Pages

GivenRVYandx.gl'Disafunction

7iscreeeiEgc.DK 2ygoDpcyx1Continuous EEgc.DKffgodfglHdyandfyylxlH

tK

iIn tncxit.at

nlwhereEExn

MbutvCxnI 9bestondardnormaldistribution