马萨诸塞州理工学院电气工程与计算机科学系.pdf

MassachusettsInstituteofTechnology

DepartmentofElectricalEngineeringComputerScience

6.041/6.431:ProbabilisticSystemsAnalysis

(Fall2010)

Tutorial3:Solutions

1.IngeneralwehavethatE[aX+bY+c]=aE[X]+bE[Y]+c.Therefore,

E[Z]=2·E[X]−3·E[Y].

Forthecaseoftrandomvariables,wehavethatifZ=a·X+b·Y,then

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var(Z)=a·var(X)+b·var(Y).

Therefore,var(Z)=4·var(X)+9·var(Y).

2.Seeonlinesolutions.

3.(a)Wecanfindcknowingthattheprobabilityoftheentiresamplespacemustequal1.

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1=XXpX,Y(x,y)

x=1y=1

=c+c+2c+2c+4c+3c+c+6c

=20c

1

Therefore,c=20.

(b)pY(2)=P3pX,Y(x,2)=2c+0+4c=6c=3.

x=110

(c)Z=YX2

E[Z|Y=2]=E[YX2|Y=2]

=E[2X2|Y=2]

=2E[X2|Y=2]

pX,Y(x,2)

pX|Y(x|2)=pY(2).

Therefore,

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