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## 3rd Midterm Study Guide

by: Michelle Schmutz

54

0

9

# 3rd Midterm Study Guide 3341

Michelle Schmutz
UTD
GPA 3.3

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## About this Document

This has the problems and things we needed to know he put extra emphasis on, and the sample exam solutions. we don't need to know #3 and #4 of the 4th question, and #6 is wrong on his solution s...
COURSE
Probability Theory and Statistics
PROF.
Dr. Mohammed Saquib
TYPE
Study Guide
PAGES
9
WORDS
CONCEPTS
probability theory and statistics, Statistics
KARMA
50 ?

## Popular in General Engineering

This 9 page Study Guide was uploaded by Michelle Schmutz on Saturday April 9, 2016. The Study Guide belongs to 3341 at University of Texas at Dallas taught by Dr. Mohammed Saquib in Winter 2016. Since its upload, it has received 54 views. For similar materials see Probability Theory and Statistics in General Engineering at University of Texas at Dallas.

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Date Created: 04/09/16
Midterm Study Guide #3 ' ' 21=1 K.090.09 0 oy ioxgz Rytx y -2 k¥ 21=2 0 0 081 ; ×=0y)y=o y0t 0 =§p×,y#y , 0.01 0.18 edpxtdxefdhffotio 10.110.040.81 Given 0'foo9 21=221=10 O 0.81 0.81 .IE?IofdoiY snEo px.ycx.gl P4#={YiliYET pyly 1- Fin: 0.o; y=. ¥¥ ;ow ,EHtEtp =§P#Ky) mPMFinal P qkntpylgh ) -2 k¥ E(D= 0*0.0-11*0.18+2*0 Bx=o y=oy㱺 y 0.01=1 xjtoo0.1/0.0910-81 :# 0.81 0.81 Pyly T Ely)=0*0.1+1*0.-12*0.8Pxytx ,y)y=oyit :0.1/0.0910-81 =1.71=0.09+1.62 y.tk#x:# 0.81 0.81 Pyly T ex Fi(w) ,where W=×y %={ 0,44 } ' ECWKEGY ) Bycx,y)y=o㱺y=z ¥÷s*¥*x . www.theikdWYYE8?oBEhEF=o*#I*o.oa+e*o.aw=X+ay 0 ; &W . 0.09+3.24=3.33 fnway±} - it'sless work wayzwMuwȼwethis technique ) evaluatethe functiofereverypossikkxoy , make E{9(×,y)}=§§g(x,yjp(×=x,y=y Matri,tpoihtwisemultiplication ' xl 0 I 2 × 't0 I 2 o 0 * 0 I 0 I 2 10.090.09 - EK ) 0010 2 4 21 00=>01*0081)+(4*0.81) 3.33 - ,y = ELW) Ecxy ) PMF joint onlyfortineas ex)Given : Learned : NEK Px ,y(x,y)=PK=X ,y=y) Find: ypx.to#taPxJxdy IF Fi a?Eule nowmebtionhebtiaship Pylyl ,ELy)=µy,Jy whenfo ,y)±EEC*D=iex,µx)(y-µD}}ch5h.314 HECXPrk .mx# formal definition lsnathexpoession g) Given: ilinldatio:ph tat iv doesn't x'sdirection y=axtb change Fin: - a=Y oppositeirections µ ,Jy ,wvkiyhpxy # X @cov(x,y)=e{(x-µ×)(y-µy )} axtb - -b) } =E{(*µ×)( aµx @ECy)=µy=aµ×tbf =E{(×-µ×X*µx)a} - aagnnocgrcoktih bOVar4Hj2y-azjx2-aEECxfuxY3-Jy-1afjfeuarisalways-a@a-opositirelsigndependsonat.aTsnowmebtichOpx.gogytxH.gITEsEfEpssFTiteoEuyawmjxt.Yna.EafYtdnttEijafdy-2dxfxY-grxXgjDgfzzcor-dx2y-2xt22x2y-Z2xz-ZJjwstusethisequationdiredlypx.y-lsY-axtb.aDiv-dx2yccovCx.ykdxdypx.y µ -h'4㱺ˠwunYneY¥hnww on +lE6y×s×gDn±H independent tlhvariannormalized byslandardeviation pt correlation tfpxyatl Correlation EGPEO coefficient orthogonal ex) Given : Xi .. ,Xnareiid㱺f× .f×n(X ,xn ) , ; . Efx ,(×?fq( xD .. .fxnK) Etfxfx ,)fxfk) .. .fxKn ) Find :P(Xn=ma×(x , ,xnDttnbstRv ,, P(Xn=minlK÷Ynȼ .. , p(Xn=max(x .,xnD ,, ¥ .ae#9oE....IonfoonfxtxDfxkd...fx(Xn-DfxCxn).x.dxz...dxn+dxn =L :( fxonofxlxm ,dxn+ ) . fxonofxlxddxz )( fhoofxlx , )d×)*f×(xn)dxn :( ) . F×lH=gP•f×(u)du=f9oEKn .tn#fxKnldxn=f9fFxcxDmfxCxn)dx.FhYTntt X ddE¥n1=fx(Xn)㱺dy=fxCxn)dxn ×nȾoo xn - too =fymdy=Info=# .am#,a..s=Etotos=*aj Foti ex)Given Ex 4 91) ,y)={2x;0IfIE㱺 h-*ȼy±xz o ;ow . y=x2 y=xz Find : . '× X=ry x±Ff Html a)C= ? ¥ I 'it i b)fx(x),fy\$ # aretheyirdeps.ho.becauseydeplndsonxinlyIExsndcovCx.yKOjHEGsymmety1DfxCxtffx.yCx.yjdyc-fyff-fry2xdx-X2lK-l-fuhenOEyLDIIIxIdjt.a.uxufetstIta2xdxax4ILttiihEHsHtsilwhmtsgtggPwauap.aqxjgooIegfttsEtEt5ooEjEY.EotBttIa.y.a.scECxkfjxtt4x3dx-4foxdx-4sXsfo-41sk9DElyHf9ooyfyly1dyoY-EiylHy1dytfloylty1dyshsx.OtrianglePDFissymmetncaound01covCx.yKFk.y2s-EWECyXo4ts-Ek.y1xtyarenotindependeutbecause4x3CktyHF2xEEgCx.yB-SxfygCx.yHx.yCX.y1dxdy-ffxyCzxdydx-2fjxyj@dx-xE2aydy-yEfxy-tzska5-fx2IB-tzszxa-x93-0-72fox2Bydydx-O-covCx.yfindependenceIsMhlYx.y ) . butcovk indep ,y)=O# ^ As ) (3,5) ex)Giren: was - Pcmaxlxykd 's dawarea =L ,y)={ okgjfjEF of interest y<w eis! -law Find: IT Square- 'x<w , ?, whereW=meX(×,y ) x,hmetobelessthanw (010) wa (310) ) ,y ( oats 't =P(XEW,YEw) PD㱺Fw(w)=P(w<w)=P(maX(x, y)Ew) •c}3) =] ffxiytx,y)dXdy w%m#nµA p ±w XEW xsw lPD㱺fw(w)= ,dslottfxyk={*o5;owtY o㱺 fame¥ sowfindiendt .gg#reas=2Fwlwdifferentti)sxdytBfx=Yls§dxdydj0Xfw/YtW 3L woes ( =3 fww.dtowwwt.to#EoEit3 㱺swpesand#r;3<w€5w%d×G㱺*y¥±wdx%y€㱺¥;o<w㱺fw(w)dw W=g;w.¥dwt\$wYsdw entire: Fw(w)=P(w< 4=1 - ) 0k ... =hP( mink , .. ) , ,×nȾiid - , ,×n)>w QW=min{ ×,...n3 =L PK , >w,Xz>w ,..,×n>w) Find: El -PCx,>w)* . .*P(xn>w ) FWW )&fw( w) =L-( tFxCwD* ...*(tF×n(WD Dense P(w>w #lttFxo..*(tFqwD subscriponx's a ' nowthesame P(Xn=Max(× ,...×n))=tn thing Fwlwfl dw fewI#xCwDnH w) )=n(tF×(wDmfx(w)f( Sample exam problems fxytxiy) texypeyjoindependent =( #2) Given: Random variablesX and 'have the PMF know :{ )-1 joint Epxycx,y fxCx7f9oofxylxifdyHDGiven.fylyHf0o0ofx.yCx.ydxRandomvaniablesXandYhavethejointCDFCtexXtett.yxE0oFxyKPszo.o.w.Findip1ugin2d3fo@aHxxe2.Yc3PFjgjyljf3HEktetFx.yCx.y-PCxLX.YfylbsFxcx.tayExsYhoEiiex-EEsoIeFEEEffyYt3tldk9deiF.TtdFylyIFx.y6o.y ,y(x,y7={ 0 ;ow . Pycyhsxpxiykyl )=teh={'€¥g EW=§xPxW a) Whetisthevalueytheconstantc Varlx)=EA2tE4D=f×2 yn 㱺 # , P(×=l)-4/28 EC×Y=EX2PxlD , ••kc Ⱦ IIF¥I⾨*¥#g*¥±a*'*Ek?stI¥D % P(Y=D -7/28 .- .co.4c Ⱦ 7128 PGH ,y=h=PkeDP(y=D=%s*s5=Yzg 1 , [ 4 >X 9 ) becauseheconditiyxincxy doesn't g pyly whoepegdedependontkconditionofy PxD= - 4128 8%8%28 3+1 -16+2+12+4=28 c)EC×)=§xP×tx) Ely)=§yPyly ) =4*IDH3*¥D =t*Is)+(z*Fsh(*¥D - . 12*758+32*224 d)j×=Fa)=EE4¥a jy ECYY =£ztYf+¥s=gI=D22*8*+42*124 = =¥s+¥s=7¥s=sD 7%8+189/28=196/28--7--21EC3=fa -4 # (1,) o :f #3) Given : Know : >x a . a., FiaTtIWrvarHEKtf0o0oxfxCHdxEtHfyH-fo5jgxyetdxioasezECx7foooxfxIddxVarlD-ECx4-E4D-f90ox2fxcxjdx-E2Wcneedthistirstcdy.l-Ek.ytElDEtDfxlDef9oofx.ylxiydy-folJdYEkytf0oof9ooxyfx.yCx.y1dxdy-tsTCxty1dy-t3CxYtEH2objEhpovarCH-t3l2xtFhyfycp-foxfyfdx-tsfloCxtyHx-tslEtyxHo-2zIt2zEH-foxtEtEsldx-JloxHkxtzDdx-tsftztyl-ItufhsSloxkxtzDdx-13floz@tzddxtHkfoyHetufDdy-st5oyktuDdy-tsKytKDfo.tk.D fx.yk.yt.ES#poeoeEEzfxhD=fFofxilKYYr=*=fxa=gjofxoET ao*e ttso2Ckty4dy-tzlKt7YDto-t3tst3D-t3k3j.tD-43.fIttsfttst3tssftsEHBvarGtEG4-E4xtf00oxzfxWdx.e2gjvarlYkECyY-E4yj-f.Yoy2fylyIdy-E2-EtyYfoy4tutttDdy-f2oHt3tyayEktfox45tHdx-fotEt25Ydx.gyotkty3jdy-t3CEthIlii5foCx3txDdx-5GtEHo-theIIeitoEl-tEtht2seaY-5.ktD-tsf3atH-3tetisteEvarlH-G-tHtEa-t2sTaFT.EvarcxFIs-Ed2-EEstHd6vCx.yHEkyhECxTEKD-5.CEXatjElxykfofoxylxtsIdxdya.sg.zgsg.qHEktYHEKHElD-Fttta-hEDtf2oyflox-yxdxdyeDVarktD-VarkItvarCptzavCxyj-Ezt3tt2tstD-tfoytEtFzIds-EztToa.aa-s@hsfoytst1DdyttzfoCfiEjdy-teEtEDi-tsCICat8D-tsaise.t , 'tneed to know how to that #6 was inthe do 3&4 ,and wrong SolutionSheet distance #4) Given : - Van choc Straw Short 0.2 0.2 0.2 oderf long 0.1 0.2 0.1 Find jointPMFPN ,p( nd) Drsaquibsaidwedon00.41+3004.3=6+40+90 ^ t op 0.2 0.1 Ⱦ 0.4 2- / - 0.2 02 0.2 Ⱦ 0.6 to do too 7 D ok ! ok bas , , D) = b)E( N Pvcn ) = 136 ^ ? f e) Are Dan independent 0.2 0.1 Ⱦ 0.4 If were,P(N=l)P(D=D should 0.2 o., they equal 2- P(N=DP(D=D - 0.2 oz Ⱦ , oz ,,Pw=D to do , = 00.18=10.2.3) too therefore Ntdaremtindepuobt ok o!q ¥ ppd - , P(D=l ) f)Cost=N*D Know : OR Elgcx ,yD=5§g(}y)p(x=X,y=y ) Usetechn .fm#2!make2matriasddopoint-wisemultipliationtaddthemup " 20100300 .it#o:::::s=OZ(Zo)+O.2(NO)+o.z( ( v zoo) 1 20 100 300 240200600=4+203+0.1400) -160+4+40+60=1880

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