Exercises II
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I I Ex€RcrsEs 8l Wc considcr thc discrctc distributior of Eramplc 25.3. Thc FMGF of X is GAt)= MArtt) = I t<2 2-t Noticc that highcr dcrivativcs arc casily obtaincd for thc FMGF, which was not thc c{s€ for thc MGF. In particular, thc tth dcrivativc is cY\t)- rt(z-tl-''t Conscqucntly, E(X) = CAI') = l(2- l)-r = l, attd EIX(X - lD = 6i(l) =2(2-l)-!=2 tt follows that E(xr)=E(x)+2=3, and thus, var(x) =3-12=L SUMMARY Thc purposc of this chaptcr was to dcvclop a mathcmatiqal structurc for cxprcss- ing a probability modcl for lhc possiblc outcomcs of an cxpcrimcnt whcn thcsc outcomcs cannot bc prcdictcd dctcrmidstically. A random variablc, which is a rcal-valucd function dcfincd on a samplc spacc, and ahc associatcd probability dcnsity function (pd0 providc a rcasonablc apptoach to assigninS probabilitics whcn thc outcomcs of an cxpcrimcnt can bc quantificd. Random variablcs oftcn can bc classificd as cithcr discrctc o! conlinuous, and thc mcthod o[ assigning probability to a rcal cvcnt ,/{ irvolvcs summing thc pdf ovcr valucs of ,.1 in thc discrctc casc, and intcgrating thc pdl ovcr lhc sct ''l ilt ahc continuous casc. Thc cumulativ. distribution function (CDF) providcs a unificd aPProach for crprcss- ing thc distribution ofp.obability to thc possiblc valucs of (hc random variablc. Thc momcnts arc sp€cial cxpccicd valucs, which includc thc mcan and variaocc as particular cascs, and also providc dcscriptivc mcasurcs for othcr charactcristi6 such as skcwncss of a distribution. Bounds lor thc probabilitics of ccrtain typ€i of cvcots can bc crprcsscd in tcrms of cxpcctcd yalucs. An important bound of this sora is givcn by thc Chcby- chcv incquality. EXEFCISES |,/ L.t. - lt,f) tcp.csent 8n {rbitrary outcom. rcsulting f.om lwo rolls of thc four-ridcd dic of Eramplc 2.1.1, Tebulatc lhc discrctc pdfsnd skclch lhc 8r.ph of thc cDF lor thc followinS random variablcsi (a) Y(.)-l+./.
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Transcript of Exercises II
IIExRcrsEs8lWcconsidcrthcdiscrctcdistributiorofEramplc25.3.ThcFMGFofX isGAt)=MArtt)= I t 3].(e) Findlhc probabilitylhatthc scorcii an odd inlctcr.3.A bag conlaintthrcccoins,oneofwhichha.sahcadonbolhsidcJ whilcthcothcrlwocointatcnormal.A coinii choscnatrandomfrom thcbagsnd torscd lhra.limcs'Th'nl,mbcrolhcadsis arandomvariablc,say X.(a) Findlhc discrct. pdfofx.(Hir!: Usc lhc LawofTolalP.obability with8r =anormalcoinand Br - two-hcadcdcoin.)(b) stctch lhc discrcl. pdf andthc CDFofX.t.A bor containr6vccolotcdballs,two blackand thrccwhilc.Balls arc drawnsucctstivclys/ithoutt.plaemcnllf X is thc numbcroldraws!ntil lhc laltblackballisobtaincd,findthc di56ctc pdtl(r}, '.r' A discrclcrandomvariablchar pdfl(x).lal lff(xl -Hll2Y forx - 1,2,3,andzctoothcrwisc,find r((b) Is afunctionoflh. foft I$l-kUIPY-ll2l fo.x-0,l,2aPdfforany,t?6.Dcnolcby [x] thc Srcalcst intcSc.no!cxcscdingx.For thc Pdfin Eramplc211,showlhatthcCDFcanbctcPrcacntedasF(x) - ([x]/12)tfor0 2].(d) Find E(xIy 8.A nonnc8ativcinlcgc.-valucdr.ndomvariabl.X ha5aCDF otthcformF(x)-I - (lPF'I for x-0, l,1,,.andzcroifx
