Definition
Let(X,Y)beatwo-dimensionalrandomvariable,foranyrealnumberx,y,abinaryfunction:
F(x,y)=P{(X<=x)cross(Y<=y)}=>P(X<=x,Y<=y)
iscalled:Thedistributionfunctionoftwo-dimensionalrandomvariables(X,Y),orthejointdistributionfunctionofrandomvariablesXandY.
ThejointdistributionfunctionofrandomvariablesXandYisthat(X,Y)isatwo-dimensionalrandomvariable.Foranyrealnumberx,y,abinaryfunction:F(x,y)=P{(X<=x)cross(Y<=y)}=>P(X<=x,Y<=y)iscalledthedistributionfunctionoftwo-dimensionalrandomvariables(X,Y).
Geometricmeaning
JointprobabilityGeometricmeaningofdistribution
Ifthetwo-dimensionalrandomvariable(X,Y)isregardedasthecoordinatesofarandompointontheplane,thenthefunctionvalueofthedistributionfunctionF(x,y)at(x,y)isthattherandompoint(X,Y)fallsonthepoint(x,y)Theprobabilitythatthevertexislocatedintheinfiniterectangularregiontothelowerleftofthepoint.
Probabilitydistribution
Jointdistribution(5photos)
Inprobabilitytheory,fortworandomvariablesXandY,ThejointdistributionistheprobabilitydistributionforXandYatthesametime.
Two-dimensionalvariable
LetEbearandomexperiment,anditssamplespaceisS={e}.SupposeX=X(e)andY=Y(e)arerandomvariablesdefinedonS.Avector(X,Y)formedbythemiscalledatwo-dimensionalrandomvectororatwo-dimensionalrandomvariable.
Discretevariables
FordiscreterandomvariablesXandY,thejointdistributionprobabilitydensityfunctionisasfollows:
.Becauseitisaprobabilitydistributionfunction,thefollowingconditionsmustbemet:
.
Continuousvariables
Similarly,forcontinuousrandomvariables,thejointdistributionprobabilitydensityfunctionisfX,Y(x,y),wherefY|X(y|x)andfX|Y(x|y)representtheconditionaldistributionofYwhenX=xandtheconditionaldistributionofXwhenY=y;fX(x)andfY(y)representthemarginaldistributionofXandY,respectively.
Similarly,becauseitisaprobabilitydistributionfunction,theremustbe:∫x∫yfX,Y(x,y)dydx=1
independentvariables
Foranyxandy,therearediscreterandomvariables:
P(X=xandY=y)=P(X=x)·P(Y=y)
Ortherearecontinuousrandomvariables:
pX,Y(x,y)=pX(x)·pY(y)
ThenXandYareindependent.
Multivariatecombination
ThebinaryjointdistributioncanbeextendedtoanymultivariatecaseX1,...,Xn
fx1,…..,Xn(x1,….,xn)=fxn∣x1,...,xn-1(xn∣x1,...,xn-1)fx1,...,xn-1(x1,...,xn-1)