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)

Tutorial10

November18/19,2010

1.DefineXastheheightinmetersofarandomlyselectedCanadian,wheretheselectionprobability

isequalforeachCanadian,anddenoteE[X]byh.Boisinterestedinestimatingh.Becausehe

issurethatnoCanadianistallerthan3meters,Bodecidestouse1.5metersasaconservative

(large)valueforthestandarddeviationofX.Toestimateh,Boaveragestheheightsofn

Canadiansthatheselectsatrandom;hedenotesthisquantitybyH.

(a)IntermsofhandBo’s1.5meterboundforthestandarddeviationofX,determinethe

expectedvalueandstandarddeviationforH.

(b)HelpBobycalculatingaminimumvalueofn(withn0)suchthatthestandarddeviation

ofBo’sestimator,H,willbelessthan0.01meters.

(c)Bowouldliketobe99%surethathisestimateiswithin5centimetersofthetrueaverage

heightofCanadians.UsingtheChebyshevinequality,calculatetheminimumvalueofn

thatwillmakeBohappy.

(d)IethatnoCanadiansaretallerthanthreemeters,whyisitcorrecttouse1.5

metersasanupperboundonthestandarddeviationforX,theheightofanyCanadian

selecteda