Quant-Job-Interview-Questions

QUANTJOBINTERVIEWQUESTIONSContents1.BasicMath12.Probability23.MarkovChainsApplications44.BrownianMotionandStochasticCalculus55.OptionPricing61.BasicMath(1)Onemorning,inSpringfield,somewhereintheUS,itstartedsnowingataheavybutconstantrate.HomerSimpsonhadjuststartedhisownsnow-plowbusiness.Hissnowplowstartedoutat8:00A.M.At9:00A.M.ithadgone2miles.By10:00A.M.ithadgone3miles.Assumingthatthesnowplowremovesaconstantvolumeofsnowperhour,determinethetimeatwhichitstartedsnowing.(2)Ahollowconeofsemi-verticalanglearctan(12)isfixedvertex-downwardswithitsaxisverticalandisbeingfilledwithwaterataconstantratek.Aspiderwhichcanrunatspeeduisasleepattheendofaweb,hangingataheighth0verticallyabovethevertex.Find,intermsofkandh0,theminimumvalueofuthatwillenablethespidertoescapeasoaking,assumingthathestartstorunupwardsassoonasthewatertoucheshisfeet.(3)LetAbeann⇥nmatrix,whichisskew-symmetric,i.e.AT=A.Ifnisodd,provethatKer(A)6=~0,whereKer(A)isthekernelorthenull-spaceofAconsistingofallvectorskilledbyA,i.e.allvsuchthatAv=0.(4)IfAisaskew-symmetricmatrix,i.e.AT=A,thenprovethatA2issymmetricandnonnegativedefinitematrix.(5)Considerafull-rankmatrixAofsizembynandavectorb2Rm.(a)Assumethatmn.Findx2Rnwiththesmallest2-normthatsolvesAx=b.(b)Assumethatmn.Findx2Rnthatminimizes||Axb||2.Date:May82009.12QUANTJOBINTERVIEWQUESTIONS(6)Let(.)bethecumulativedistributionfunctionofaN(0,1)randomvari-able.Defineafunctiong(x)asg(x)=Zx2lnx(t+x)dt.Evaluateg0(1)intermsof(.),whereg0(x)isthefirstderivativeofg(x).(7)Acubeoficemeltswithoutchangingshapeatuniformrate4cm3/min.Findtherateofchangeofthesurfaceareaofthecubewhenthevolumeis125cm3.(8)Solvetheequationforx2R:sx+rx+qx+p...=x(9)Findywherey=11+11+11+1...2.Probability(1)aandbarerandomlychosenrealnumbersintheinterval[0,1],thatisbothaandbarestandarduniformrandomvariables.Findtheprobabilitythatthequadraticequationx2+ax+b=0hasrealsolutions.(2)Whatistheexpectednumberof(fair)coinflipstogettwoconsecutiveheads?Hint:Computeexpectationbyconditioning.(3)Youaretrappedinadarkcavewiththreeindistinguishableexitsonthewalls.Oneoftheexitstakesyou3hourstotravelandtakesyououtside.Oneoftheotherexitstakes1hourtotravelandtheothertakes2hours,butbothdropyoubackintheoriginalcavethroughtheceiling,whichisunreachablefromthefloorofthecave.Youhavenowayofmarkingwhichexitsyouhaveattempted.Whatistheexpectedtimeittakesforyoutogetoutside?(4)Consideralog-normalrandomvariableXgivenbyX=em+sZ,whereZisastandard-normalrandomvariable,mandsarerealconstants,andsispositive.QUANTJOBINTERVIEWQUESTIONS3(a)ComputetheprobabilitydensityfunctionofX.(b)Computemean,median,andmodeofthedistributionofX.(5)SupposeXisaGaussianvariableofmean0,andstandarddeviation,anddefineN(u)astheCDFofacenteredGaussianvariableofmean0andvariance1,i.e,N(u)=1p2⇡Zu1es22ds.(a)CalculatetheexpectationofN(X).(b)CalculatetheexpectationofN(X+a),wherea0.(6)Bysettingupandsolvingtheappropriateconvolution,showthatthesumoftwoindependentzero-meanunit-varianceGaussiandistributionsisGauss-ian,withvariance2.(7)Ifyourepeatedlyflipacoinwhoseprobabilityofheadsispthenwhatistheexpectednumberofflipsyouneedtodoinordertogetaheadimmediatelyfollowedbyatail?(8)SupposeXisarandomvariablewithE[X2]1.WhatistheconstantcthatminimizesE[(Xc)2]?(9)LetXbeacontinuousrandomvariablesuchthatP(X0)=0.Letf(·)beitsprobabilitydensityfunction.Supposethatf(x)/P(Xx)isequaltoaconstantforallx0.WhatisthedistributionfunctionofX?(10)LetfX(·)bethepdfofacontinuousr.v.X.WhatisthepdfofeX?(11)Youhave1000coins,oneofwhichisfaulty:ithasaheadonbothsides.Yourandomlydrawacoin,and,withoutexaminingit,tossit10times.Asithappens,youget10headsinarow.What’stheprobabilitythatit’sthefaultyone?(12)LetXbearandomvariablewithdensitydistributionfunctionf(X).DefineF(x)=Pr[Xx].WhatisE[F(X)]?(13)Assumethearrivaltime,T,ofacreditdefaulteventfollowsanexponentialdistributionsuchthatP(Tt)=1ehtfort0,andP(Tt)=0fort0.Thequantityh0iscalledthedefaultrate.Whatistheexpectedvalue,E[T],ofthedefaultevent?(14)Consideragamewhereweareallowedtothrowadice3times,andwewinanamountequaltothenumberonthedice.Weareallowedtostopwhenwewantandobtainthevaluecorrespondingtothelastthrow,butifwe4QUANTJOBINTERVIEWQUESTIONSthrowit3times,wemustacceptthelastvalue.Howmuchisthisgameworth?(15)AbugcrawlsalongtheedgesofaregulartetrahedronABCDwithedgeslength1.ItstartsatAandateachvertexchoosesitsnextedgeatrandom(soithasa13chanceofgoingbackalongtheedgeitcameon,anda13chanceofgoingalongeachoftheothertwo).Findtheprobabilitythatafterithascrawledadistance7itisagainatA.(16)Anantandablindspiderareonoppositecornersofacube.Theantisstationaryandthespidermovesatrandomfromonecornertoanotheralongtheedgesonly.Whatistheexpectednumberofturnsbeforethespiderreachestheant?(17)Twoarchersshootatatarget.Thedistanceofeachshotfromthecenterofthetargetisuniformlydistributedfrom0to1,independentlyoftheothershot.WhatisthePDF(probabilitydensityfunction)ofthedistancefromthecenterforthewinningshot?(18)Theheightofmen(ininches)intheUSisapproximatelynormallydis-tributedwithmean68inchesandstandarddeviationof5inches.(a)Adoorframemanufacturerwantst