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# Week 4 - Probability Theory and Statistics 3341

UTD

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This 6 page Class Notes was uploaded by Michelle Schmutz on Saturday February 13, 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 26 views. For similar materials see Probability Theory and Statistics in General Engineering at University of Texas at Dallas.

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Date Created: 02/13/16

Weekly #4 Notes 8+10,2016 Probabilitytion February Binomiali Recapcwhatwelearnedinch2) Probabilityunhon P(X=x) Experiments -Samplepacei) resuygnasbimwyew Possofaneveuteithessuueedingorsailiy S={Si,,5n muloutcomes . § }±P Coiflip S={HIT} < x=o x=i HȾO ȼ P S×={}1 basicafunction TȾ1 c-1- y=×bothyhanru C.3Ylmimkk(RD= Rv(DRD{eVed J×ȾP×(x)=P(X=x) fnmx 㱺€PxH=#;PxN>€ PC)±€.P(X=x) S(X=D Hhesucceats h'll someth,ng ##+# Giv:nlosesiFin:ayitsdum6Hoftests-1-n@5xandnotevenworthhistin.bepxcxHhpYxiG.stp.xosen.P .at/mys#qtIgXpi= T nti9YEirtsktY onsito -o㱺Pk㱺㱺Hx㱺 @Sk P ={ # poitbelire 0 ; ..1(P(X=l)㱺SCx=1) #. Bernoulli ; otherwise Binomial Given Untilhsemeststa,tse's k=#Itoftsts howayofkuowigfrswuhow of Success(s of htesfs manyteststherewillbeinbinomid i=.jp#gnksk=o.i.....oFnoPttD P÷spout ten,tsn } test Find: 0 P(K=¥)=P( 0 . ! succ"soutqntests 0,1,,n} =n @PkH#ȼH={ Geometric Pascal Geometric Given # of trials needed by the $6 milliogambling man to succeed onthefirsttest 4 Tz Ts , t.IE?pot Come ON ,I KNOW , . , ,Ty I CAN WIN THIS TIME ! 0¥ GEOMETRIC Find: " THESE , .IM#YM&nfinite ## DKE 1BOUGHT ARE @ ={ ,,}, & SUPPOSED ToBE EXTRA gy Man wins ! @py(y)=P(Y=y) )=P=CtpDP=( LUCKY Py (1)-P(y=1)=P( ta"P PyH=PCy=y=PCF£)=PA㱺P(H=( 1- ) Py(3)=P(y=3)=PCFiES3)=PCFDPA 㱺P(S})=¢P3÷ Pyly ltptisry .1,2, ... )=P(y=y)={ 0 Ow . ; TH ex Eambliy Man has an addictionto gamblingd won'tgo home untilhe wins X times. IF1 WIN4 MORE TIMES I'LLBE ABLE TO BUY , Given: THAT trialsneeded to a §%°YF+M"%RGHwh . it'llE Green Z=#q get successes AND ILL NAMEIT AFTER THE CHAMELEON IN TANGLED Find : - tobasiaxy NAMED PASCAL ! @ P Sz={x , at J,xP ,..} inanity @ +2×+3 , Pz¢D=PzKD=P( getting successesithinZ. tria)s = PAAB ) = PAM D= PAI =P tenuity . ' ' PH )= PK -) successouof E-1trialsȾ A- (2-1) Sz infrst'z -Itrials = z÷toy .sn#y=IiiQayMAD**PY*oH*ajmoYytp.i*aa+i.. When the the ex) mailman scans apackage, thetracker uploads information so customers can getupdates on the Status the:# .The chances itmaking an errorWith probabilityindependent othercorrectly of package of p, of any updated package a) Ifthe takes keepsupdatingcorrectlyntilit receivestsirsterrorhat ithe Probabilityss Function(PMF) U , of , the number of updates ? Puke ) aberration 's )=p(U=u ȼ P§Yu÷t§÷F¥ek= Pepped . atop 0 OW mgsotnseigdsto ; PK ,CzeD=P¢DPCa)P(ed=(tp¥ : correct terror . :If p= 0.1,whatisthe probabilitytU=N ? Given " =D P(U=lo)=(tDnP=dpP9p =(1.0 0.15×0.1=0.9*1 dozjtenhaveto What is the probabilitythat Uz 10? Given : p=¥ Uzlo ) =p u=N$u=H$u=r$ ) ( ... =P (4=10 )+P(U=1DtP(U=Ht ...becausthey'rmutually exclusive factortry =(-p9Pt(1- )Pt .. pjnpt(tp tumblr.TL?QEnteghf=f.p)9EPtHoPHhP5Pt..j 㱺{P(u=Dtp(u㱺)tP(u=3)t . 㱺{P+CtD ¥1 'tKPTPT ...L} =p Pfa Dow # C) Ifthe tracker updates 100 what isthe PMF Y, the number incorrectupdated ? - packages , of of packages Pylyhptky ) ( ( ; a = . n { O on ; ftp..pk?Hoss=as...#o d) Ifthe more state. the -art truckers and the chances error reduce down to 0.01 post office got of of , what isthe probability 4=3 errors - ) = of Py (g) Ptky ( log)P4hp)9s What is the ' probability of Yes errors PC KD =P ('to $Y=1 UUU't 2*4=3 ) 9 ' ' remember =P ( to )tp(Y=1)+P( HHPN =3) its = " I PH-D 's He)* mutually (1000)p°(Huot ( 10)0 pytpj + ( :o) +¢o)P3( exclusive e) H the until the 's 3 What isthe the rank tracker Keeps updating post offices computes receives errors, PMFQZ , number packages updated ? of Given - # of trialsto 3 errors get '' 3 ' Pzfz (Q ) PYHD ; 2=3,4 ,,... p z= 2)=O )=PCz=zD={ 0 ; Ow - ( f) If p -0.25 , what isthe probability E- R packages updated until the computer of receives 3 eras ? P( -p) 9 = ' z=W=¢)p3a fly3(tPy㱺 A rangya sample space has all of the possible kountablxphysial outcomes range Cumulativdistribuproperty Poisson Px ,1,2 :a > 'Nt{o ;X=o ,range 0 TI ;ow . ex) Given : .pe#I=EoejgagF.ix=Q1.z Rat isproportionalo T㱺Xt=# constant XFX ED Some birds are sitting on a telephone wire . so can be on the wire at a time it starts Only many before getting cramped . If two more birds arrive what is the probabilityhat 5 will leave? , P(Bz=s ) The the 10- Poisson ED number of updated packages processed by postoffices computer in any Second intervalis a random Variable S ,L , with a- updates What isthe that there be in 10-second interval? $ probability will no updates processed a Given : Find : ¥10 sec P (no updates in EIO seconds =P (× s ,x=5 ,o=o)=et*€#[ e- X ,o=X10=>52 oj x=÷÷ What isthe probabilitythat attest two updates willbe processed in a second interval * PC 2 updates in F- 2 sec TPKZZZ ) = 1- PCKEO ) .PK#1)=teaokI . eȾk÷ ! =te¥ -@ For a discrete random variableX with PMFPXKD and Sx : range a) For x any , P×k)z0 b) Exesxpxiy 't c)For event BE Sx the that X in the set B is PCB ) any , probability is £ȾPx(x)=£㱺PCX=D Given : Find : Pxlxtpkx ) - Plotpxa } } } ; * * ={?§gxEe b) P(x€ -0.00004=0 ###¥n#I = * . :O£ £ 0. w . C) PCXEO ) 314 added theproperty d) P ( XE Of9991=3/4 CumulativeistributivectiKDF ) anxetto = e) P(x{ /) = f) p( X 1100 ) 1/4+3/4=14=2 . g) PCXEO Property Mass Function 㱺 Cumulative Distributiveunction D Fxfx ) - Pke -5=0 2) Fx too )=P(×< too) =L DX >¥' ' (XDZFXK ) Only when increasingrremainingonstan(non decreasingfunctio)CCDA 4) Xie Sx Jump occurs when there's a value inside the qx range Fx (X Fx ( - E) = - E a ;) X ; PXCXDTPG XD , where an arbitrary very small positiveNumber €0 5) that area remains flat Xicxlxi it FxW=F×Cx ;)

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