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1、FRM-级培训项目一.3T相比率非每件W雅住二2个工具FDFC斤三I，公为贝叩斯TermsRanomexperiment（gfl1Anobservationormeasurementprocesswithmultiplebutuncertainoutcomes.Outcome（STheresultofasingletrial.Forexample,ifwerolltwodices,anoutcomemightbe3and4;adifferentoutcomemightbe5and2EVent停件）1表列CUtaPme（结果）的集台Theresultthatreflectsnone,one,or
2、moreoutcomesinthesamplespace.EventsCanbesimpleorcompound.Aneventisasubsetofthesampleace.IfverolltwodicesiaexampleOfaneventmightberolling7in2【M1.Mutuallyexclusiveevents?件）.Eventsthatcannotbothhappenatrhesametime/gg曲eegO15彘各事件）ThoseindudeMPGSSibgProbabilityrobAbrHfyE11-,*”，.*IT*lIMff.U,rm1lITlf,.，n，1p
3、(Z)=F(A同乂RB)StctP门Theprobabilitythattherandomvariables(inthiscase,bothPIA玲三P(Al麻randomvariables)takeoncertainVakJesimultaneous电P(AB).同吁也UnconditionalProbability(边际概窣,akaInargi咆七吧也n*力TheexpectedValUeOfthevariab/ewithoutanyrestrictions(orlackinganypriorinformation),P（八）.ConditionalProbability僚件概率)GAne
4、xpectedVaIUeforthevariableconditionalonpriorinformationorsomerestriction(e.g.,thevalueofacorrelatedvarrabJe).TheconditionalexpectationofB,conditionalonA,isgivenbyP(BA).ProbabilityandProbabilityDistributionsI需ionalpJCSi收磕篇嘉森侬,HJwJAQonditionaUaobabiHtPAB)J-)僦iT触翅邢P(AR)P(BM)=灰亦P:O今ProbabilityandProbabi
5、lityDistributionskJointlirlji.ibtlity;f*(0jFVJ升中MJ我5一Mult.(,l.cjtonnk+、W*PmCrS)=Pm)，P(B)广俨8)IfAandBarcuIu!IyjjcU51yc.evcnt$the11：/P(AorB)=P（八）+P(B)门A-BA旷PSB)二ProbabiHtyandProbabilityDistributionslr0evens狭娜斤斤学互稼倚，TheoccurrenceofAhasnoinfluenceIndependenceandMutuallyExclusivearequitedifferent.HPXrhjGi
6、V户mjqnctjdfnrndc力方百/CauseexclusivemeansifAoccur.Bcannotoccur,AinfluentsBstrongly.Probability0ProbabilityandProbabilityDistributions三ITk汽Hfj1隋件才崎kTota(robabilityFormufa1LA2力贝叶期。IfaneventAmustresultinoneofthfcmutuallyexclusiveeventsApA,fAj,.,An.then1P（八）=P(AJP(A丹J+P(A?)P(AlA9+.P(An)P(AlAn)二.登也)也庇恒飞是砌先
7、篦AiAiH/)UtI4=%AfeNC)二Ply+晔=P（八）NqA)十&Wq朗合ProbabilityandProbabilityDistributionsAExample任一卷词J央叮砒我吵祕M斤十%.L三IIProbabilityandProbabilityDistributions可喇m,%,$)?慢份病ruB)E想个啮叮呻脚学HJrHMrhruP(AlB)令ProbabilityAlgorithmIS)HalfofthemortgagesinaPortfdioareconsideredsubprime.TheprincipalbalanceQfhalfOftheSUbPrimemor
8、tgagesandOne-quart沸Ofthenon-subprimebmortgagesceedsthvalueo、21介赤*画丽selectamortgagefromtheportfolioforreviewanditsPiDugancfifxcued5tlgY曲史Ofthecollatewh*stheprobabilitythatitisajumgmortgage?1/41/3仙号尸K1/2/2/3ACorrectAnswer:Dbc令巾-SibrtJ匕&中5.lI一一一rrrm=aJ，“ProbabilityFunction&CumulativeDistnbutioriFunctio
9、nProbability令ProbabilityandProbabilityDistributionsProbabiHDtbDescribetheprobabilitiesofallthepossibleoutcomesforarandomvariable,Threebasicforms:/Diagram.；/TableormatrixMnctifi115弓Px=k)=其中=,2345bProbabilityandProbabilityDistributionsA，”JVkv尸ptscrrlrdConiniuoiHri011VrtriflblcsD5rcterjndr11,-.1.tcIhrn
10、umhu腐%.Olulim11pr.h11rt!r*flIj11fJIorr.*,hpo*.Mb*Htforrm,th.nu.ur.b(randpos11vrprobbhtyConhnuouvv.11,tlr.Hipr11Hnhttofpntjr()tc3ruk*.匕bir.rvrnifIosvrrandupperboundsevsiP、-OCMCBIhoughXcanoccurgjXVqmteiW尸g%9令 Probability and Probability DistributionsI 二21XAT 5 2)adenk Probability function: p(x) = P(X
11、) FQr discrete random VariableS旭林与S -P(X) = 1P(XT)P(X-5).川Q-/Probability dentity funci6n抱f:,(x)X X* X。Forlcontinuousfrandom variable commonly CUEUl口tivc probability fuctiQ (aJ; f(xo Discrete Jht;耀牛房缺，)= P(X V 幻* Continuous/ Fa) = J二/()dM)XlC吁做今ProbabilityandProbabilityDistributionsARandomVariablesan
12、dTherrProbabilityDistributionsProbabilityDistributionOfa0isceteRandomVariable)/ProbabifrtyMassFunction(PMF)orProbabilityFunction(PF)f(Xxi)二P(X-Gi=1.2,3；，PropertiesofthePMF令f(X=Jq)=xH.令0f国)1)=l令ProbabilityandProbabilityDistributionsProbability Distribution Of a Continuous Random Variable ProbebiUty d
13、CnSity RJVCtion (PDF) A PDF has the following properties:/ The total area under the curve f(x) is 1/ P(xj X x2)- P(Xl X x2) = P(Xl v X V 工2) to AI ItUnrT3afiue DTtribUtiOn FUrlcion (CDF) 令ProbabilityandProbabilityDistributionsPropertiesofCDFF-=andF+J=Q丹的耸我力皿eF(X)isaR11jf萩虱functionsuchthatifx?x1thenF
14、(x,)闵F(XJF、,P(Xk)=l-F(k)灾P(XIX42)=F(Xz)-F的)；今 Probability and Probability Distributions Ml4AttPrChnhililydEJiy functi” Pdj)Lw少Ie We lake X from 1 or 2 WlthIhUamenrQu)bity We take YP(Aft)-FlA)Y产吗产7IQtJl Pl10254-A0.7S 20.25 I0 25n仁口and Y=9 50100hofn Xl vth tfe snjSl rW Properties Of HlQ bivariate or j
15、oint probability mass function_(PMF)r 引尸P(X=LaW)。卜PCY) Ve for all pairs of X and YThiS is because all probabilities are nonnegMiVa lf(,) =1PIXHI)=0 5令 Probability and Probability DistributionsMarginal probability distribution Of X and YVaIueofXfx)Value of Y电）10.501075205020.25Sum1.00 ILOO MMgi“I Pro
16、bability function Definition of marginal probability functionKX) = Tf(X1Y)InrallXf(Y) = f(X,Y)furaljr令 Probability and Probability Distributions Statistical Independence23fY)11/91/93/921/91/91/93/931/91/91/93/93/93/93/91 Definiti OnofStatiStiCalP【XH 用引)二HXyXKYN)!第烦？一I独卷完前借的所Rca;是图i.1ProbabfIHy】dProb
17、abilityDistributionsPdrir*,.1Vv.ibn.1八,“，*I；I*1I1l*JM4I，hh，h，/!hLtI!1*3V“K -不便,亦P：X Jtml ho /”，IlnCd (re I twP.-1-=-；.-*-f=一=rUExpectedValueandVarianceBasicStatistics摘述性统mf舍CentraLMoment&kMomc,*ThCkjhmomentOfXisdehncdasm,E(Xf)6；!卜LUiCiitftiXo1iUu11jUUr二_一_kCentralmoment7hrklce11!,tJnuFntaHl3，dtlfdat
18、*Ut*Xr；if卫thefirstcentralmomentisequaltoO.日”)Ul-?Thftrfcn4jnt3mcnen!仁IhEIdCanCEd=E(Xj!汪二力henhcjlrdjnl4jrcube】lk =力 JhcN工史幺辽W EDnl空l3VidPd by计比Cfth。standarddeviationisHleHJnaWKT芹squareOfthevarianceisthejkjrl5s7lE寿令ExpectedValue1.ExpcctedVdIuc中/昆争Ameasureofcentraltendency-thefirstmoment.E(X)MSP(xi)Xj=
19、P(X1)X1+P(切*P(QME(X)Jxf(x)dx 加才又含ExpectedValue尸PropertiesofExpectedValue9KblSaE(b=b&Ifais3c115ta11L厄aX)=aE&Meifaandbareconstants,thenEqX+b)=a日X)+Eb)二。E(X?)川E(X)H受郎7磷数时革：(K另sE(X+Y)=E(X)*E(YJ).#sIngeneral,E(XY)WE(X)E(Y);IfXandYar电延画歪血独立randomvariables,thenjE(XVE(X)E(YF侬丈PH)tF*)若J版则P(AB)=N今*网令Variance问
20、UMlmw川(hsrronIhrvrcifnHf11G*)/IX“At_、，八川.1P-,Hu*ijfrtr(fv,fhrur11puththr*vt,*,v.v.tHfnrrhFF/CX-)M.V)-。Mkasunf.h(11vIinrsyorrr(lctjbcOutrandorrvarMbl?11H*p0Sdvequjrcootu力；，疗,.方Vnvn己帛theF竺CJKUgriMg口aspC3ldtot11iT21VarianceCr (X El 6吵) 二，E(X9 二 d%)尸PropertiesofVarianceHcconstant,then:7-O.Ifd&CflDSianit
21、hen：qaX)二a?。Xx).IfblSaConStanLthen:o?(X+b)=J(X).制 J2(X) = E(X?) TE(Xn7 NX+D= f 尸!fgj少LttagJgg11Lthen:o?(aX*b)=a2?XL十IfXandYareSampMeanThesamplemeanofarandomvariable,X,isdefinedas:斤二Zrr为hlThesamplemeanisknownasanestimatorofE(X)1whichwecannowcallthepopulationmean.Anestimateofthepopulationissimplythenu
22、mericalvalue令SampleVariance7序 TheSampIe u)t. lcntt1W oni111isknownasherkqrgcofjerd!白由藤Cxplainthed.f.(degreesoffreedom)。ttrfirpwprnmp1ptheSfCMmhSnrPV.,vr*mustfcomputeS.RutWICeweUSethesampsampletocomputeX.haw5八6n,mdependeMobsevabonstocomputeS;.sotospeak,weloseIdI.SampleMeanandVariancePopulationSampleM
23、ean时N平VarianceFi1滔StandardDeviation回;8Chebyshev,sInequalityChCbyshevsIneqUalityForanySetofobservations(samplesopopulation),theproportionOfthevaluesthatliewithinkstandarddeviationsofthemeanisatleast1-l/N.k1.P(IX-W也i)目二ThkrelationshipappieseeoLhpfieUbg#YA尸色厂2CorrelationCoefficientEK IlThankyou!Correla
24、tion-ToMficierit BaSiC Statistics W相岩嘴C两组金罪令 Covariance7 O 产 SiMZe=O 而.etSnsyPropertiesofCovariance.IfXandYareindependentrandomvariablestheircovarianceiszero.jsyih4XPU0 Cov (XlX)=El(X-E(X)(X-E(X)I=Qi(X)%V(OXOX)=吟lXK)&力也吵工0KW(X3)#Therelationshipbetweencovarianceandvariance-”布2(XY)-(X)+i(Y)2Cov(X.Y)吁1
25、*e(MHhhdrnfthew(t)v.UhC7rtFlirHMorrr5P(O11rl.1nncrhcirrlvllbr“。TlirnrmIh*uvi5EUtranpfeVaE(j)叫=EaI)Wl+E(j)叫+E(X3)HE(Xn)%3N个)=尸借UARt):玄唱自吗SMR硬阈)2(RP)=WWWj叫COV(Ri,RD* COy(XJJ=右丫死皆1=1JlmarketvalueofimuestmcntinassetiWj=marketualueoftheportfolioNa=/(叫R+以山)、_(tv,P)f(L)r-2-yaBpa-*._*,.*_BagSjbCoskewnessand
26、卜CokurtosisBasicStatisticsSkewness注：Ada伙brttMyossetrehims如t力,Skewness(三河Ameasureof-asymmetryofaPDF一thethirdmoment,刃田g吧h吵骨尸抬尻巧吁OIrtOTU姓笠心级更/t伽E(X-)3thirdcentralmomentaboutmeanS=，=.F一3cubeofstandarddeviationSymmetricalandHonsymmetricaldistributionsPositivelyskewed(rightskewed)andnegativelyskewed(left夕
27、呻瓦ZnaHl相喷旗捣n在用电厂Negattve-SkeweoSymmetncPositive-skewedMean=Median=ModeW*核PositiveskewedMod?制直/mean,havingaghtfattailexMt)侬设massto电比呼寸Ule0严(向v，ueNegativeskewed:Modemediamean,havingaleftfattail令KUrtoSiSUptokdttie mtourtie pftyt(iMic II二!BAJM 4CGJI LH 1*l n,r,nj irq I. fJP (1(, * K v.ft R. 7e / R. Sf*K
28、Riskmodelswithtime-varyingvolatilityortime-varyingcorrelationcandisplayawiderangeofbehaviorsWIthveryfewfreeprmeters4ihi-tail8沱Hi5(庙豳联性)aCopulascanalsobeusedtodescribecomplexinteractionsbetweenvariablesthatgobeyondcovariances,andhavebecomePoPUlarinriskmanagementinrecentyears.3CoskewnessandCokurtosisE
29、xampleAumcfourscriesofhindrclurn5(A.氏&D)whorethemean,standarddcvalin.skew,andku11oisarealltheampjbutonlytheordcfofctussdifferentUmeIAonr.DCDIilliriCA*日COIK3”153%I1ereturnsQflhetw。portfoliosaredifferent,isbecausetheCQSkCWne55betweentheportfoliosisdifferent.Notices,毛尸。GandF依andD)碱,w一0S8”099Qf3ttWQVa11
30、aDieS.)d!U/Forexample力灯=EK胃尸说明辞手无处游，9Ja)=EIX-讲IL()yS=日X-EX)，疝ThenontrivialCpkurlgsisCrftwoVanaMes：(and玉，/ForexampleKXXxY二SmWn终=tyj令BestLinearUnbiasedEstimator(BLUE)川口3grBLUlnolhrrproprr1yof1111(jfjlIhliphouiiiHfrhr114tr(*,itc；“itjrusdas.lf%MhMhcn11ltl11r.a)erlata)Utheestimator5theh，J.v(ablrk(1.h%lh),(xhfl
