概率论与数理统计公式Probability theory and mathematical statistics formula.docx
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概率论与数理统计公式Probabilitytheoryandmathematicalstatisticsformula
概率论与数理统计公式(Probabilitytheoryandmathematicalstatisticsformula)
Thefirstchapterstochasticeventsandtheirprobability
(1)permutationsandcombinationsformulasareusedtopickoutthepossiblenumberofpermutationsofnindividualsfrommindividuals.
Thenumberofpossiblecombinationsofnindividualsselectedfrommindividuals.
(2)additionandmultiplicationprincipleadditionprinciple(twomethodscancompletethematter):
m+n
Thetwomethodcanbeusedtocompleteacertainsubject.ThefirstmethodcanbecompletedbyMmethods,andthesecondmethodcanbecompletedbynmethods.Then,thismethodcanbecompletedbym+nmethods.
Multiplicationprinciple(twostepscannotdothisseparately):
mxn
ThefirststepcanbecompletedbyMmethods,andthesecondstepcanbecompletedbynmethods,andthiscanbeaccomplishedbym*nmethodsintwoways.
(3)somecommonpermutationsarerepetitiveandnonrepetitive(ordered)
Oppositeevents(atleastone)
Orderproblem
(4)randomtestandrandomeventsifatestisrepeatedinthesameconditions,andeachtimethetestresultsmaybemorethanone,butbeforeatestisnottoassertthatitappearswhichresults,saidthestudywasarandomizedtrial.
Thepossibleoutcomeoftheexperimentiscalledarandomevent.
(5)basicevents,samplespacesandeventsinatest,regardlessofthenumberofevents,canalwaysfindsuchagroupofevents,ithasthefollowingproperties:
Eachtrialmustoccurandonlyoneeventinthisgroupoccurs;
Anyeventismadeupofsomeoftheeventsinthisgroup.
Eacheventinsuchagroupofeventsiscalledabasicevent,whichisusedtorepresenttheevent.
Thewholeofthebasiceventiscalledthesamplespaceofthetest.
Aneventisacollectionofparts(basicevents)init.CapitallettersA,B,Careusuallyused,...Representingevents,theyaresubsets.
Isitaninevitableevent,animpossibleone?
.
Theprobabilityofanimpossibleeventiszero,andtheeventwithzeroprobabilityisnotnecessarilyanimpossibleevent;similarly,theprobabilityoftheinevitableevent(omega)is1,andtheeventwithprobability1isnotnecessarilyaninevitableevent.
(6)therelationshipbetweeneventsandoperations:
IfthecomponentoftheeventAisalsoapartoftheeventB,(Ahappens,theremustbeaneventB):
Ifthereisasimultaneousevent,theeventAisequivalenttotheeventB,orAequalsB:
A=B.
ThereisatleastoneeventinAandB:
AB,orA+B.
AneventthatispartofAratherthanBiscalledthedifferencebetweenAandB,denotedasA-B,andcanalsobedenotedasA-AB,oritrepresentstheeventthatBdoesnothappenwhenAoccurs.
AandBoccursimultaneously:
A,B,orAB.AB=?
whichmeansthatAandBcannothappenatthesametime,calledeventAincompatiblewitheventBormutuallyexclusive.Basiceventsareincompatible.
-AiscalledtheinverseeventofeventA,ortheoppositeeventofA.ItrepresentsaneventthatdoesnotoccurinA.Mutualexclusionisnotnecessarilyopposite.
Operations:
Bindingrate:
A(BC)=(AB)C,A(B,C)=(A,B),C
Thedistributionrate(AB),C=(A,C)a(B,C)(A,B)C=(AC),(BC)
Therateofprobability:
(7)theaxiomaticdefinitionofprobabilityissetasasamplespace.Forevents,thereisarealnumberP(A)foreachevent,ifthefollowingthreeconditionsaresatisfied:
1~0=P(A=1),
2degreeP(omega)=1
3degreesfor22incompatibleevents,,...Yes
Itisoftencalledcountable(complete)additivity.
P(A)iscalledtheprobabilityofevents.
(8)theclassicalprobabilitymodelis1degrees,
2degree.
Setanyevent,itismadeupof,thereis
P(A)==
(9)geometricprobabilityiftherandomtestresultsforinfiniteuncountableandeachresultsthepossibilityofuniform,andeverybasiceventinthesamplespacecanbeusedtodescribeaboundedregion,saidthetestforrandomgeometricprobability.Aforanyevent,
.Lisgeometricmeasure(length,area,volume).(10)additiveformulaP(A+B)=P(A)+P(B)-P(AB)
WhenP(AB)=0,P(A+B)=P(A)+P(B)
(11)subtractionformulaP(A-B)=P(A)-P(AB)
WhenBA,P(A-B)=P(A)-P(B)
WhenA=,P()=1-P(B)(12)conditionalprobabilitydefinesAandBaretwoevents,andP(A)>0iscalledtheconditionalprobabilityofeventBoccurringineventA.
Conditionalprobabilityisakindofprobability,andallprobabilitypropertiesaresuitableforconditionalprobability.
Forexample,P(omega/B)=1P(/A)=1-P(B/A)(13)multiplicationformulamultiplicationformula:
Moregenerally,foreventA1,A2,...An,ifP(A1A2...An-1)>0,butthereis
…………。
(14)independence:
theindependenceofthetwoevents
Eventandsatisfactionarecalledevents,andtheyareindependentofeachother.
Ifeventsaremutuallyindependent,andthenthereare
Ifeventsareindependentofeachother,theycanbeseparatedfromeachother.
Inevitableeventsandimpossibleeventsareindependentofanyevent.
Ismutuallyexclusivetoanyevent.
Theindependenceofmultipleevents
LetABCbethreeevents,if22independentconditionsaresatisfied,
P(AB)=P(A)P(B);P(BC)=P(B)P(C);P(CA)=P(C)P(A)
AndsatisfyP(ABC)=P(A)P(B)P(C)atthesametime
SoA,B,Careindependentofeachother.
Similartonevents.
(15)allprobabilityformulasetseventsatisfaction
1degrees22incompatibleeachother,,
2degree,
Isthere
.
(16)Biasformulasetevent,,...Andsatisfaction
1degree,,...22incompatible,>0,1,2,...,,
2degree,,
be
I=1,2,...N.
ThisformulaistheBayesformula.
(,,...Aprioriprobability.(,,...Itisusuallycalledposteriorprobability.TheBiasformulareflectsthe"causalprobabilitylaw",and"madebyShuoyinfruit"inference.
(17)wehavedoneatestonBernoulli'shypothesis
Eachtrialhasonlytwopossibleoutcomesthatoccurordonotoccur;
Thesecondarytestisrepeated,i.e.,theprobabilityofoccurrenceishomogeneousateachtime;
Eachtrialisindependent,thatis,whethereachtrialoccursornotdoesnotaffecttheoccurrenceofothertrials.
ThisexperimentiscalledtheBernoullimodel,ortheheavyBernoullitest.
TheprobabilityofoccurrenceofeachtrialisexpressedastheprobabilityoftheoccurrenceofthesecondintheheavyBernoullitest,
,。
Thesecondchapterrandomvariableanditsdistribution
(1)theprobabilityofdiscreterandomvariableisXk(k=1,2),...Andtaketheprobabilityofeachvalue,thatis,theprobabilityoftheevent(X=Xk)is
P(X=xk)=pk,k=1,2,...,
Theupperboundistheprobabilitydistributionordistributionlawofdiscreterandomvariables.Itissometimesgivenintheformofdistributedcolumns:
.
Obviously,thedistributionlawshouldmeetthefollowingconditions:
(1),
(2).
(2)thedistributiondensityofcontinuousrandomvariablesisthedistributionfunctionofrandomvariables.Ifthereisanonnegativefunction,thereisanarbitraryrealnumber
Itiscalledcontinuousrandomvariable.Probabilitydensityfunctionordensityfunction,referredtoasprobabilitydensity.
Thedensityfunctionhasthefollowing4properties:
1degree.
2degree.
(3)therelationshipbetweendiscreteandcontinuousrandomvariables
Thefunctionoftheintegralelementinthetheoryofcontinuousrandomvariableissimilartothatinthetheoryofdiscreterandomvariables.
(4)thedistributionfunctionisarandomvariable,anditisanarbitraryrealnumber
ThedistributionfunctionofX,arandomvariable,isessentiallyacumulativefunction.
YoucangettheprobabilitythatXfallsintotherange.Thedistributionfunctionrepresentstheprobabilityoftherandomvariablefallingintotheinterval(-,x]).
Thedistributionfunctionhasthefollowingproperties:
1degree;
2degreesaremonotonedecreasingfunctions;
3degree,;
4degrees,thatis,rightcontinuous;
5degree.
Fordiscreterandomvariables,;
Forcontinuousrandomvariables,.(5)eightdistribution,0-1distributionP(X=1)=p,P(X=0)=q
IntheNouritest,thetwodistributionistheprobabilityofeventoccurrence.Thenumberofeventsisarandomvariable,andifitis,itmaybevaluedas.
Amongthem,
Itiscalledthetwodistributionofrandomvariablesobeyingtheparameter.Rememberas.
Atthattime,thisis(0-1)distribution,so(0-1)thedistributionisaspecialcaseofthetwodistribution.
ThedistributionlawofrandomvariablesisgivenbyPoissondistribution
,,,
ThePoissondistribution,whichiscalledtherandomparameter,isdenotedasorP().
ThePoissondistributionisthelimitdistributionofthetwoterms(np=,N,P).
Hypergeometricdistribution
ThehypergeometricdistributionoftherandomvariableXfollowstheparametern,N,M,denotedbyH(n,N,M).
Thegeometricdistribution,whereinP=0,q=1-p.
ThegeometricdistributionoftherandomvariableXobeyingtheparameterpisdenotedasG(P).
Thevalueoftherandomvariableisonly[a,b],andthedensityfunctionisconstanton[aandb]
Other,
Therandomvariableisuniformlydistributedon[aandb],andisdenotedasX~U(a,B).
Thedistributionfunctionis
Whena=x1.exponentialdistribution
Amongthem,theexponentialdistributionoftherandomvariableXobeystheparameter.
ThedistributionfunctionofXis
Remembertheintegralformula:
Thedensityfunctionofrandomvariablesisnormaldistribution
,,
Wheretheconstantiscalledtherandomvariable,thenormaldistributionorGauss(Gauss)distributionisassumedastheparameter.
Ithasthefollowingproperties:
Thefigureof1degreesisaboutsymmetry;
At2degrees,themaximumvaluewasthen;
If,thenthedistributionfunctionis
..
Thenormaldistri