概率论与数理统计英文题目

Test

1.Considerthesetwithsubsets

Findthefollowingsets:

(a)and(b)and

2.ShowDeMorgansLaw:(a),(b)

3.(a)LetP(A)=0.5,P(B)=0.4,P(A∩B)=0.2.FindP(A∪B)andP(A|B)

(b)ConsidertwofairdiceAandB.DieAissix-sidedandisnumbered1throughto6whilstdieBisfour-sidedandisnumbered1through4.Bothdicearerolled.Findtheprobabilityoftwodiceshowthesamescore.

4.(a).LetXbearandomvariable..Showthat

(b)ThepmfofarandomvariableXwhichhasaPoissondistributionwithparameter2.FindP(X=3).

5.Suppose(X,Y)bebivariatenormaldistribution.

.FindthecorrelationcoefficientofXandY.

6.(a)LetXUniformlydistributedon(-1,1),i.e.X~U(-1,1).FindtheExpectationandVarianceofX.

(b)LetXandYbecontinuousrandomvariableswithjointpdfLetZ=X+Y,Find.Inparticular,ifXandYareindependent,FindfX*fY..

7.(a)TherandomvariableXhaspmfis:X0123

PX(x)0.20.160.410.23

FindP(1≤X<3)andFX(1.7).

(b)LetXbeacontinuousrandomvariablewithpdf

FindP(X=3),P(X>1)and.

8.Xiscontinuousr.v.withpdf

Let,FindE(Y)

9.LetXbeacontinuousrandomvariablewithprobabilitydensityfunction

(a)FindP(1<X<2)(b)Find:(1)E(X);,(2)GiventhatE(X2)=6.FindVar(X)

10．Ifthejointprobabilitydensityfunctionoftworandomvariablesisgivenby

(a)FindthemarginalpdfofXandY.

(b)Determinewhetherthetworandomvariablesaredependentorindependent.

11．SupposetherandomvariableXhasthedensityfunction

FindtheprobabilitydensityfunctionoftherandomvariableY=X+4.

12.Takeoutanumberfrom1-200(200naturalnumbers).

Find(a)theprobabilitythatthisnumbercanbedrivedby6;

(b)theprobabilitythatthisnumbercanbedrivedby6aswellas8.

13.Threechildrenareselectedatrandomfromagroupoffiveboysandthreegirls.

(a)Whatistheprobabilitythatallthreeareboys?

(b)Whatistheprobabilityatleasttwogirlsareselected?

14.(a)Let

(b)Let，，.Find.

15.(a)SupposeA,Baretwoevents,andP(A)=1/4,P(B)=1/2,P(AB)=1/9,thenevaluate

(b)UseBayes’TheoremtoshowthatifP(A),P(B)>0andthen

16.(a)LetXhasaBinomialdistributionwithparametersnandp,i.e.X~b(n,p).Find

(b)Showthat

17.SupposethedistributionfunctionofarandomvariableXis

，

Find(a)theprobabilitythatXgetsvaluewithin(0.3，0.7);

(b)thedensityfunctionofX

18.TheoperationallifetimeX,inyears,ofabatterypoweredwatchhasprobabilitydensityfunction

(a)Findthevalueofc.

(b)FindthecumulativedistributionfunctionofX.

(c)Findtheprobabilitythatthewatchhasanoperationallifetimeinexcessof4years.

19.(a)TherandomvariableXhastheprobabilitymassfunctionbelow:

X

1

2

3

4

Px(x)

0.4

0.14

a

0.24

Findaand

(b)AcontinuousrandomvariableXhavingtheprobabilitydensityfunction.FindandP(0<X<1).

20.(a)Let.FindP(AB)

(b)SupposeXandYareindependentrandomvariables,andFind

21.Supposethedensityfunctionof(X,Y)is

(a)Determinetheconstantk.(b)FindtheprobabilityP{X<1,Y<3}.

22.LetXdenotethenumberoftimesacertainnumericalcontrolmachinewillmalfunction:1,2,or3timesonanygivenday.LetYdenotethenumberoftimesatechnicianiscalledonanemergencycall.TheirjointprobabilitydistributionisgivenasTable1.1

Table1.1

X

Y123

10.050.050.1

20.050.10.35

300.20.1

(a)EvaluatethemarginaldistributionofXandY

(b)DeterminewhetherthetworandomvariablesofXandYaredependentorindependent.

23.(a)If,thenfind

(b)Let(X,Y)be2-dimensionalrandomvariables,and

If，find.

24.Supposetheprobabilitydistributionofisasfollows,and.

FindthecorrelationcoefficientofXandY.

25.LetXhasabeuniformdistributionontheinterval(0,1),and

Find(a)(b)E(Y)

26.TherandomvariableX,forfixed0<p<1andn≥1,hasprobabilitymassfunction(pmf)

(a)Findcintermsofpandn.

(b)FindthecumulativedistributionfunctionofX.

27.If，,XandYareindependent.Find.

28.Thelengthoftime,inminutes,foranairplanetoobtainclearancefortakeoffatacertainairportisarandomvariableY=3X-2,whereXhasthedensityfunction

FindthemeanandvarianceoftherandomvariableY.

29.SupposetheprobabilitydensityfunctionofrandomvariableXis

,

and.

Find(a)theprobabilitydensityfunctionofY,.

(b)expectationE(Y)andvarianceVar(Y).

30.