Mathematical Statistics with Applications, 7 edition REVISED_AISE_Chapter04

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1、Chapter4:ContinuousVariablesandTheirProbabilityDistributions?0y<1?.41≤y<2??4.1a.F(y)=P(Y≤y)=?.72≤y<3?.93≤y<4???1y≥4F(y)0.81.60.40.20.00.0012345yb.Thegraphisabove.4.2a.p(1)=.2,p(2)=(1/4)4/5=.2,p(3)=(1/3)(3/4)(4/5)=2.,p(4)=.2,p(5)=.2.?0y<1?.21≤y<2??.42≤y<3b.F(y)=P(Y≤y)=?.63≤y<4??.84≤y<5??

2、1y≥5c.P(Y<3)=F(2)=.4,P(Y≤3)=.6,P(Y=3)=p(3)=.2d.No,sinceYisadiscreterandomvariable.Chapter4:ContinuousVariablesandTheirProbabilityDistributions60Instructor’sSolutionsManualq=1-pF(y)0101y4.3a.Thegraphisabove.b.Itiseasilyshownthatallthreepropertieshold.4.4Abinomialvariablewithn=1hastheBern

3、oullidistribution.4.5Fory=2,3,…,F(y)–F(y–1)=P(Y≤y)–P(Y≤y–1)=P(Y=y)=p(y).Also,F(1)=P(Y≤1)=P(Y=1)=p(1).sti4.6a.F(i)=P(Y≤i)=1–P(Y>i)=1–P(1itrialsarefailures)=1–q.b.Itiseasilyshownthatallthreepropertieshold.4.7a.P(2≤Y<5)=P(Y≤4)–P(Y≤1)=.967–.376=0.591P(2

4、adiscretevariable,sotheyarenotequal.b.P(2≤Y≤5)=P(Y≤5)–P(Y≤1)=.994–.376=0.618P(2

5、≤1)=.648.c.Sameaspartb.above.d.P(Y≤.4

6、Y≤.8)=P(Y≤.4)/P(Y≤.8)=.352/.896=0.393.e.Sameaspartd.above.Chapter4:ContinuousVariablesandTheirProbabilityDistributions61Instructor’sSolutionsManual4.9a.YisadiscreterandomvariablebecauseF(y)isnotacontinuousfunction.Also,thesetofpossiblevaluesofYrepre

7、sentsacountableset.b.Thesevaluesare2,2.5,4,5.5,6,and7.c.p(2)=1/8,p(2.5)=3/16–1/8=1/16,p(4)=1/2–3/16=5/16,p(5.5)=5/8–1/2=1/8,p(6)=11/16–5/8=1/16,p(7)=1–11/16=5/16.d.P(Y≤φ)=F(φ)=.5,soφ=4..5.5.5φ.954.10a.F(φ)=6y(1?y)dy=.95,soφ=0.865..95∫.950b.SinceYisacontinuousrandomvariable,y0=φ=0.865..9

8、52224.11a.cydy=[]cy/2=2c=1,soc=1/2.∫00yy2tyb.F(y)=f(t)dt=dt=,0≤y≤4.∫∫24?∞00.00.20.40.60.81.00.00.51.01.52.0yc.Solidline:f(y);dashedline:F(y)d.P(1≤Y≤2)=F(2)–F(1)=1–.25=.75.e.NotethatP(1≤Y≤2)=1–P(0≤Y<1).Theregion(0≤y<1)formsatriangle(inthedensitygraphabove)withabaseof1andaheighto