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# Week 6 - Probability Theory and Statistics 3341

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This 8 page Class Notes was uploaded by Michelle Schmutz on Thursday February 25, 2016. The Class Notes belongs to 3341 at University of Texas at Dallas taught by Dr. Mohammed Saquib in Winter 2016. Since its upload, it has received 28 views. For similar materials see Probability Theory and Statistics in General Engineering at University of Texas at Dallas.

## Reviews for Week 6 - Probability Theory and Statistics

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Date Created: 02/25/16

Weekly Notes #6 Monday 2122¥ : Recap toget ) PM PFXE , now you Pylyt ,§#=Pyxlx only need gsumst to PMFY get Ely ) EH )=,€,9KB# now youdon't even need PMFY , only need gk)Px(X) E(aXtb)=aEWtb If y=ax+b ,Ely )=E{ axtb} =aE(×)tb ron 'tneedPMFX either g) When the postman scans packages the scanner updatesthe package's location but an errorcan occur The , . postoffice system can these.When an updateis receiveerorfee , the System sends a (VD back confirmation tothe scannerWhenthe system receives error, a negation (NOK ) message is sent back to the Sannes . Each time that is the scanner receives an NOK , the update retransmi,s. Itis assumed each update independentlycorrupted by errorswith each consists probabilityof. update Collect of multiple bits bib T*y##€;*#g÷r× oinfnup ' receives 'ed¥ not a) Find: bits corrector. # PMF×( oftimes the scanner updatesapackage ) ×=2 delay Tx ) , TH - . .- - #µmerw - x=z #f%µY¥#¥#sd hsotgtginhessP FDPCH #%1 pk=D=q°H 8) %1d.II.lk#.=82tt8 ' ' Had ¥0 Wto ix. =8" a FEi#x)={F' ( H ;x=k Geometrics 0 . transmits untilfirst ; OW Success Know : Geometric RV has EGD =tp 㱺 So Ek)= € b) Suppose that each update takes I millisecondused to transmittothe system, and the scanner waits another Imsec to from to receive a OK or NOK the system before trying send the update again .Let Tbethelime required untilthe is What isthe relationshipbetween Tandx . What is the ? update Successfully . PMFOFT ¥4 ) Ⱦ The more× spend the Moret willhave . tgk you , you t.FI#IIKD1Kitx=zrxEkQTggxins@inseat=2*fIpEEFtIsar . ma ma Yett =3msec㱺T =3 why -1 ?? ftp.gxehi ;2.-Msecittookforupdates ZFTT totransmitfosystem T=g(×)=§× -| sendbactheacknowledgement PMFH ) PCx=H={8*' 㱺Pt#=Pt=D ¥g¥¥¥#e㱺-Ⱦµ#Hp;tt ST={ 1,3/5,.:} , Pt=t)=p(zxt=t ) =Pkx=tTD ,}s ... 0 ttlz .z=t÷ ; Ow . Et )=2Ek)t =3 -1 to ' Ⱦ y={ X -EKB if know admired RV found know you y , , you thinformatontfatitcarnies . 㱺#)="{ ( x-EkD2 } TaveeagevariationqRVwithrespecttoEHIVarWeDStudenttsaveragetimespeutfomgoinghomehomUTD.ExperimeDTtEtD-5minutEotalwaysthcasgitcansometimestakelonser-VartD-C9minutesTDr.SaquibbtriplengthExpoimen@TeHts.r 25 E ftp..IS minutes 'E%toT mtD=2sminzD Yadon . =㱺FM# - =S Cansometimes square this moreconstruct,To tyoeidgett) traff,c Standard & accidents deviatio(2) Kinsky Thestudentspendsonfmimtestogdhome Son , average . 0 IS 20 Drisaquib 10-20 minutes togethome 15 on . spends , average F#t= W Smin Smin cangetthvarianuwhenyouhave EG ) you because rightnow ,we cancalculatevalnesbutthat hotwhatwewanttodo right now. Goal : Toderinean standard deviation Vaalipa)=E{(×=EkD2} .la#=22xp " {*Ek)5]㱺[( :[ xz.ge#+E4xB=EkY.EE2xECxBtEEE4D } constant Ⱦ E(ax)=aEk) # =Ek272E{x}rEk)tE4D =EC×D2E4×)tE4x- ) Y=gK) Var(×)=ECx2Ek2)=qx2PCȼ㱺⾨Ely)=ElgCxD=×£gk)Pk=㱺 =Ekz¥÷ Calculationof used butgiwsttleformation important giinformation ex) Given Find: Experimented× EG ,Vark) PxlD=K⾨D Ek)= -1*+2+1*1=0 ± tz Vark)=Ek4-E2KT° xtitoto 's =E: HY*C㱺+EI* 's =£+£ =L e×)Experimeut#Y ¥ =O Given: Find f)=Etp2+ElyTattDat+tH* Ely,Va)ly Vary at at Eta =#pz outcomeof variance t'rt } %eamqusgeegd.gr#5*tattD4aDtGD4H+G25F=1+lH+W+1 =# Learned: 0Hy=axtb Ely )=aECDtb , tolearn: G2g lfy=ax+b ,Varly)= ? Vary) means Eely -Ely =E{(a¥tbHa±Wtb5} =E{ ( axhb.at#x)H23=EElaxaEkD9=EEa2CxFtxD9=a2EKx.ElxD3=a2varg~Vank)0VarCyka2varK) # , Find )Elyrly) ⊥r(y)=a2varQ)rak# |EW EW=-S =Ef± (1×1)=1 =tD2varCx ) thenyatinedidn'tchange = therariancesinaitwas Squared )=Et×)tb constantsthatyoaadddoesn't =- ECDTS thevariance the = - Sts affect only Ely= multiplieres O Wednesday February 24,016 Ch .3 recap Eat ,§xPk=X) Px - ± } 'N=Pk=X) VARKHEKD E4I⾨ , 2PK㱺 ±㱺 "' ' ,¥ trh .4 ÷ood slime ¥ 10:00am arrivingime Experiment toclass Whatisthe ¥. probabilitye student will < , arrive @ exactly 10:00am ? -S stone , E- t=+S (9:55am) (10:00)m (10:05)m because time of offs,t Yonan'tpinpointe So Pttfdoesntexigt - exactomenttheStudent CDF PtH=P(T =O)=O enterstheroomatexactg 10:00.00 Ftfb )=PGȼ6)=0 ȼ arrivedeitherexactly ogfptfhtkyoeasq µ . Aging'd $ 0¥00'Ttinbhn mpartbxumayu $ FfD=ptȼD=o insibefore EG$=P(TȼtD=1 woo:X this isbasically impossiblehryouto:O0:00:00 aontinuousfundion mnerealizeexactlyatlo Efttptct ) 1 ⊥ Continuous RV 㱺 no jumps . 1 a >>t tStt this Xisa characteristicRV - Fx(x)=PC×<X) t QF×foo)=O 2 FXGOD - +1 3 PCXILXEXZ ) OFXCXZIFXLXD Toucan differentiatcontinuousunctions £ ) - fxcx) Fxcx *Lu)du dldx DerivedRV t# - speed velocity acceleration t.ua ) + alsogettxcx PH=§gPK=xst¥n PCBH ,f€sE¥§ almostthsamaspa XD Given: : Find fxfx) F×W=P(x< 4) =%of×(x)d× Tf %%u >x UB recap Ch.4 ) =) basicalreplacing Ek)=£xPC⾨X Ektftoooxfxcxdx E⊥kD={gHP(×=x) 㱺 Ecgkkftooglxifxadx } +µFfwwiTd§pf eitherCDFORPMF willbegiveninblems eDGiven: Fin: Fyty=P(y<y) a)P(yȼD=FyH=O d)P(y>1s)=1PCYEID ^ b)P(yEtD=FytD=at sty=2㱺F =L. ¥ ) cseek 2<413- =tz=§ 4 2=3/4 > =Fy(3) Fyk ) e)P(y<D=P(µ)-p(y=aT° yttytoyttiyit -2/4 =FytD =Yq =Y4 A ' f# -makessense eDfy(y)=dgFyC$={ okjnoouhit Yaginuma na y=o nfyfgl e×)Ely)=f°oyfy(y)dy Yg 㱺/os4Wdy=f!¥dy % , so y=e >y 㱺 'gyY!=l6p=D 22=4 eDVarlyKECyY-E4yT-VarCyfk3-qaElyadooy2fyffdytbtt2g-fd-ftyitxkdy-tafoy2dyFzxx43-l6zfo0umeudu-9FfF.Cr-D.1whenrisproportioualtoapositiveintergest9eDGiven.faxxifxaD-yXExYxzoexpqgwial0.O.w.xEEbkYeamhsx-ohxxgtimportantRv.ui.rminmiviN.8EongMgeegYgatEmfwDFindxkD-PCxE2DEH-f0o0oxfxCx7dxLfxx-u-Eoofxwdn2gamnnf-sooxfKxdxfE@EefygEE0oIuEtooa-S00xXexxdx1d-So0xxExxdx0oooI0oKIfneeEItooareauntil-teixx.xzo.fo00ueiudgx-o-If0fu2iiudu-lxtxMD-txC2tYvaras.t i ¥ Ekz)=f%x2fxK)dx =D Ⱦ='s =f°oox2f×tDdX =f%ox2xe*dx = .tt#I**Vwkt*nngiemggrgaea Yu =±f°x2x2E*dx =¥fooCxx5e*dx iasooutkueu = # ÷! 㱺 ÷

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