概率论与数理统计公式Probability theory and mathematical statistics formula.docx

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