MATH3801/3901Classtest2-4May2012-QuestionsQuestionsforstudentsinMATH38011.Thedispatcheratacentralrestationhasobservedthatthetimeb etweencallsisanexp...

MATH3801/3901Classtest2-4May2012-QuestionsQuestionsforstudentsinMATH38011.Thedispatcheratacentralrestationhasobservedthatthetimeb etweencallsisanexp onentialrandomvariablewithameanof32minutes.a)Itisnowno onandthemostrecentcallcameinat11.35AM.Whatistheprobabilitythatthenextcallwillarriveb efore12.30PM?b)Whatistheprobabilitythattherewillb eexactlytwocallsduringthenexthour?2.Sho ckso ccurtoasystemaccordingtoaPoissonpro cessofrate?.Supp osethatthesystemsurviveseachsho ckwithprobabilitya?[0,1],indep endentlyofothersho cks.Whatistheprobabilitythatthesystemissurvivingattimet >0?3.Jacklikestogoshing.Whilewaitingfortheshestobite,heformulatesthefollowingmo delforthepro cess:shesbiteaccordingtoaPoissonpro cesswithintensity4bitesp erhour.Bitingshesarecaughtindep endently,andonaverageonlyoneintwotimes.a)Whatistheprobabilitythatsixshesbiteduringthersttwohours?b)Whatistheprobabilitythathefailstocatchanyshesduringthersttwohours?c)Whatistheprobabilitythat,duringthersttwohours,sixshesbiteandtwoofthesearecaught?4.CustomersenterastoreaccordingtoaPoissonpro cessofrate?= 5p erhour.Inde-p endently,eachcustomerbuyssomethingwithprobabilityp= 0.8andleaveswithoutmakingapurchasewithprobabilityq= 1-p= 0.2.Eachcustomerbuyingsomethingwillsp endanamountofmoneyuniformlydistributedb etween$1and$101(indep en-dentlyofthepurchasesoftheothercustomers).Whatarethemeanandthestandarddeviationofthetotalamountofmoneysp entbycustomerswithinanygiven10-hourday?5.Menandwomenenterasup ermarketaccordingtoindep endentPoissonpro cesseshav-ingresp ectiveratesoftwoandfourp erminute.Itisno onandtherearecurrently10customersinthesup ermarket.Fromnowon,whatistheprobabilitythatatleasttwomenarriveb eforethesecondwomanarrives?QuestionsforstudentsinMATH39016.Let{N(t),t=0}b eaPoissonpro cessofrate?.Fors,t >0,determinetheconditionaldistributionofN(t)giventhatN(t+s) =n.Whichdistributionisthis?Interpret.7.Aradioamateurwishestotransmitamessage.ThefrequencyonwhichshesendstheMorsesignalsissub jecttorandomdisturbancesaccordingtoaPoissonpro cesswithintensity?p ersecond.Inordertosucceedwiththetransmission,sheneedsatimep erio dofasecondswithoutdisturbances.Shestopsasso onassheisdone.LetTb ethetotaltimerequiredtonish.DetermineE(T).(Hint:byparts,??xe-?xdx=-xe-?x-1?e-?x)
8.Let{N1(t);t=0}b eaPoissonpro cessofrate?.Assumethatthearrivalsfromthispro cessareswitchedonandobyarrivalsfromasecondindep endentPoissonpro cess{N2(t);t=0}ofrate?.Let{NA(t);t=0}b etheswitchedpro cess,thatis,NA(t)includesthearrivalsfrom{N1(t);t=0}duringp erio dswhenN2(t)isevenandexcludesthearrivalsfrom{N1(t);t=0}whileN2(t)iso dd(seediagramb elow).a)Giventhattherstarrivalinthesecondpro cesso ccursattimet,ndtheconditionalprobabilitymassfunctionforthenumb erofarrivalsintherstpro cessuptot;b)Findthe(unconditional)probabilitymassfunctionforthenumb erofarrivalsintherstpro cess,{N1(t);t=0},duringtherstp erio dwhentheswitchison;(Hint:?80xne-axdx=n!an+1)c)Giventhatthenumb erofarrivalsoftherstpro cess,uptotherstarrivalinthesecondpro cess,isn,ndtheprobabilitydensityforthetimeofthatrstarrivalinthesecondpro cess.9.Jacklikestogoshing.Whilewaitingfortheshestobite,heformulatesthefollowingmo delforthepro cess:shesbiteaccordingtoaPoissonpro cesswithintensity4bitesp erhour.Bitingshesarecaughtindep endently,andonaverageonlyoneintwotimes.a)Whatistheprobabilitythatsixshesbiteduringthersttwohours?b)Whatistheprobabilitythathefailstocatchanyshesduringthersttwohours?c)Whatistheprobabilitythat,duringthersttwohours,sixshesbiteandtwoofthesearecaught?10.BusesarriveatacertainstopaccordingtoaPoissonpro cesswithrate?.Ifyoutakethebusfromthatstopthenittakesatimer,measuredfromthetimeatwhichyouenterthebus,toarrivehome.Ifyouwalkfromthebusstoptheittakesatimewtoarrivehome.Supp osethatyourp olicywhenarrivingatthebusstopistowaituptoatimes,andifabushasnotyetarrivedbythattimethenyouwalkhome.a)Computetheexp ectedtimefromwhenyouarriveatthebusstopuntilyoureachhome.(Hint:byparts,??xe-?xdx=-xe-?x-1?e-?x)b)Showthatifw 1/?+rthenitisminimizesbylettings=8(thatis,youcontinuetowaitforthebus);andwhenw= 1/?+rallvaluesofsgivethesameexp ectedtime.c)Giveanintuitiveexplanationofwhyweneedonlyconsiderthecasess= 0ands=8whenminimizingtheexp ectedtime.2