Mathematical Statistics

Transcription

1 ahmaical Saisics 4 Cha IV Disc Disibuios Th obabili modls fo adom ims ha ill b dscibd i his ad chas occu ful i alicaios Coiuous disibuios ill b sd i cha This cha ill ioduc som disc disibuios icludig Boulli disibuio biomial disibuio Poisso disibuio gomic disibuio gaiv biomial disibuio ad hgomic disibuio 4 Boulli Tials ad Boulli Disibuios O a sigl ial of a im suos ha h a ol o vs of is sa c ad Fo aml ad c could s h occuc of a had o a ail o a sigl coi oss obaiig a dfciv o a good im h daig a sigl im fom a maufacud lo o i gal succss o failu o a aicula ial of a im Suos ha occus ih obabili P ad cosul c occus ih c obabili P A adom vaiabl ha assums ol h valu o is o as a Boulli vaiabl ad a fomac of a im ih ol o s of oucoms is calld a Boulli ial I c aicula if a im ca sul ol i succss o failu h h cosodig Boulli vaiabl is if c if Th mf of is giv b f ad f Th cosodig disibuio is o as a Boulli disibuio ih h aam ad is mf ca b ssd as f I is obvious ha h obabiliis sum o o f f f Poof: Thom 4-: If is a adom vaiabl ih a Boulli disibuio h µ σ ad < < µ f f f f f f [ ] σ f f f < <

2 ahmaical Saisics 4 4 Biomial Disibuio Of i is ossibl o sucu a mo comlicad im as a suc of idd Boulli ials h h uai of is is h umb of succsss o a cai umbs of ials I a suc of idd Boulli ials ih obabili of succss ad obabili of failu o ach ial l s h umb of succsss Th disc mf of o as h biomial disibuio ad dod b ~ B is giv b f Th uaiis of ad a calld h aams of h biomial disibuio o ha b h biomial hom h obabiliis sum o o; ha is f aml 4-: I is o ha scs oducd b a cai coma ill b dfciv ih obabili iddl of ach oh Th coma slls h scs i acag of ad offs a mo-bac guaa ha a mos of h scs is dfciv Wha ooio of acags sold mus h coma lac? Soluio: If is h umb of dfciv scs i a acag h is a biomial adom vaiabl ih aams Hc h obabili ha a acag ill hav o b lacd 9 is P P Hc ol 4 c of h acag ill hav o b lacd Summaizig a biomial im saisfis h folloig ois: A Boulli succss-failu im is fomd ims Th ials a idd Th obabili of succss o ach ial is a cosa ; h obabili of failu is 4 Th adom vaiabl cous h umb of succsss i h ials Thom 4-: If is a adom vaiabl ih a biomial disibuio ih aam ad ~ B h µ σ ad < < Poof: f l

3 ahmaical Saisics 4 [ ] l [ ] [ ] [ ] [ ] σ < < aml 4-: A commuicaio ssm cosiss of comos ach of hich ill iddl fucio ih obabili Th oal ssm ill b abl o oa ffcivl if a las o-half of is comos fucio Fo ha valu of is a -como ssm mo lil o oa ffcivl ha -como ssm? Soluio: As h umb of fucioig comos is a biomial adom vaiabl ih aams i follos ha h obabili ha a -como ssm ill b ffciv is 4 4 has h cosodig obabili fo a -como ssm is Hc h -como ssm is b if 4 hich ducs o o

4 ahmaical Saisics 44 4 Gomic Disibuio ad gaiv Biomial Disibuio W u o o h oblm of obsvig a suc of Boulli ials uil acl succsss occu h is a fid osiiv ig L h adom vaiabl do h umb of ials dd o obsv h h succss Tha is is h ial umb o hich h h succss is obsvd B h mulilicaio ul of obabiliis h mf of sa f uals h oduc of h obabili of obaiig acl succsss i h fis ials ad h obabili of a succss o h ial Thus h mf of is f W sa ha has a gaiv biomial disibuio dod b ~ B ih h aams ad W fis discuss his oblm h Tha is cosid a suc of Boulli ials ih obabili of succss This suc is obsvd uil h fis succss occus W sa ha has a gomic Pascal disibuio dod b ~ GO ih h aam sic h mf cosiss of ms of a gomic sis aml f I ca b vifid ha h obabiliis sum o o aml 4-: Suos ha h obabili of gi malfucio duig a -hou iod is Fid h obabili ha a giv gi ill suviv hous Soluio: Lig do h umb of -hou ivals uil uil h fis malfucio hav Psuviv hous P 964 Thom 4-: If is a adom vaiabl ih a gomic disibuio ih aam h µ σ ad < Poof: To fid h ma ad h vaiac fo h gomic disibuio ill us h folloig suls abou h sum ad h fis ad scod divaivs of a gomic sis Fo < < l g a a

5 ahmaical Saisics 4 Th g a a ad g a a f σ [ ] [ ] < Thom 4-: P > Poof: P > Thom 4-: o-mo Po If ~ GO h P > j > j P > h j ad a ogaiv igs Poof: Lf as cis aml 4-: Cosid h oblm of obaiig a adom odig of h fis osiiv igs Fo aml suos ha ould li a adom odig of h fis osiiv igs W ould obai his adom odig b ollig a fai si-sidd di Th fis cas of h di ould giv h fis oucom i h adom odig To obai h scod umb i h odig ol fiv of h si ossibl oucoms a ligibl Af h fis osiios hav b filld ih uiu igs h umb of cadidas fo osiio is 6 fo 4 6 Th obabili of slcig o of hs ligibl igs is 6 6 If dos h umb of ials dd o obsv h fis succss a ig ha has o b slcd h has a gomic disibuio ih A isig oblm is o dmi h avag umb of cass of h di o obai a adom odig of 4 6 L ual h umb of cass uid o fill osiio If l W 4 6 h W dos h oal umb of cass

6 ahmaical Saisics 46 uid So h avag umb of cass uid is 6 W L us o u o h gaiv biomial disibuio Th aso fo callig his h gaiv biomial disibuio is h folloig Cosid h h biomial ih h gaiv o Usig aclaui s sis asio hav h If l i h summaio h ad h summad of hich is c fo h faco h gaiv biomial obabili h Accodig h abov fomula obai [ ] f Thom 4-4: If is a adom vaiabl ih a gaiv biomial disibuio ih aams ad B ~ h µ σ ad < Poof: To fid h ma ad vaiac ill us h folloig divaivs of h sis asio of h gaiv biomial L h Th h ad h

7 ahmaical Saisics 47 L ] [ L ] [ ] [ Va Va σ < aml 4-: Tam A las am B i a sv-gam sis Tha is h sis is ov h ih am is fou gams Fo ach gam PA is 6 ad h gams a assumd idd Wha is h obabili ha h sis ill d i acl si gams? Soluio: L b h umb of gams i h sis Th 6 B4 ~ if am A is h sis ad 4 B4 ~ if am B is h sis Hc hav is sis i 6 4 A P is sis i 6 4 B P sis gos 6 gams P Th gaiv biomial oblm is somims fd o as ivs biomial samlig Suos ha B ~ ad B ~ W I follos ha P W P

8 ahmaical Saisics 48 Tha is W cosodig o h v of havig o mo succsss i ials ad ha mas o f ials ill b dd o obai h fis succsss o ha h umb of ims i h biomial im is a fid umb hil i is a adom vaiabl i boh h gomic ad h gaiv biomial disibuios W summaiz h chaacisics of hs h disibuios i Tabl 4- Tabl 4- Biomial Dis Gomic Dis gaiv Biomial Dis T of ims Idd Boulli Tials ih h Pobabili of Succss umb of Succsss Radom Vaiabl Fid umb Fid umb umb of ims Fid umb Radom Vaiabl Radom Vaiabl

9 ahmaical Saisics Poisso Disibuio Som ims sul i couig h umb of ims aicula vs occu i giv ims o o giv hsical objcs Fo aml could cou h umb of ho calls aivig a a sichboad b 9 ad A h umb of flas i f of i h umb of cusoms ha aiv a a ic ido b oo ad P o h umb of dfcs i a -foo oll of alumium sc ha is f id ach cou ca b lood uo as a adom vaiabl associad ih a aoima Poisso ocss ovidd h codiio i Dfiiio 4- a saisfid Dfiiio 44-: L h umb of chags ha occu i a giv coiuous ival b coud W hav a aoima Poisso ocss ih aam > if h folloig a saisfid: i Th umbs of chags occuig i o-ovlaig ivals a idd ii Th obabili of acl o chag i a sufficil sho ival of lgh h is aoimal h iii Th obabili of o o mo chags i a sufficil sho ival is ssiall zo Rma I his dfiiio hav modifid h usual uims of a Poisso ocss b usig h ods aoima Poisso ocss ad ssiall i ii ad iii i od o avoid som advacd mahmaics Hc f o his as h aoima Poisso ocss Suos ha a im saisfis h h ois of a aoima Poisso ocss L do h umb of chags i a ival of lgh h lgh ss o ui of h uai ud cosidaio W ould li o fid a aoimaio fo P h is a ogaiv ig To achiv his aiio h ui ival io subivals of ual lgh If is sufficil lag shall aoima h obabili ha chags occu i his ui ival b fidig h obabili ha o chag occus i ach of acl of hs subivals Th obabili of o occuig i a o subival of lgh is aoimal b codiio ii Po chag occus i a subival Th obabili of o o mo chags i a o subival is ssiall zo b codiio iii Pmo ha o chag occus i a subival Hc Po chags occu i a subival Cosid h occuc o ooccuc of a chag i ach subival as a Boulli ial B codiio i hav a suc of Boulli ials ih obabili aoima ual o Thus a aoimaio fo P is giv b h biomial obabili

10 ahmaical Saisics 4 If icass ihou boud hav ha lim lim o fo fid hav lim lim lim ad lim Thus lim P Th disibuio of obabili associad ih his ocss has a scial am W sa ha h adom vaiabl has a Poisso disibuio ih h aam > dod b POI ~ if is mf is of fom f I is as o s ha f jos h ois of a mf sic clal f ad fom h aclaui s sis asio of hav f Th Poisso obabili disibuio as ioducd b S D Poisso i 87 This adom vaiabl has a mdous ag of alicaios i divs aas sic i ma b usd as a aoimaio fo a biomial adom vaiabl ih aams h is lag ad is small ough so ha is of a moda siz I oh ods if idd ials ach of hich suls i a succss ih obabili a fomd h h is lag ad small ough o ma moda h umb of succsss occuig is aoimal a Poisso adom vaiabl ih aam Som amls of adom vaiabls ha usuall ob h Poisso obabili la follo: Th umb of misis o a ag o a gou of ags of a boo Th umb of ol i a commui livig o as of ag Th umb of og lho umbs ha a diald i a da

11 ahmaical Saisics 4 4 Th umb of acag of dog biscuis sold i a aicula so ach da Th umb of ahuas occuig duig som fid im sa 6 Th umb of as a 7 Th umb of dah i a giv iod of im of h olicholds of a lif isuac coma aml 44-: Suos ha h obabili ha a im oducd b a cai machi ill b dfciv is Fid h obabili ha a saml of ims ill coai a mos dfciv im Soluio: Th dsid obabili is has h Poisso aoimaio ilds h valu 78 Thom 44-: If is a adom vaiabl ih a Poisso disibuio ih aam h { } ad σ µ Poof: L [ ] L [ ] [ ] σ { }

12 ahmaical Saisics 4 4 Hgomic Disibuio Suos a oulaio o collcio cosiss of a fii umb of ims sa ad h a ims of ad maiig ims a of Suos ha ims a da a adom ihou lacm ad do b h umb of ims of ha a da Th disc mf of calld h hgomic disibuio ih aams ad ad dod b HYP ~ is giv b f h is a ogaiv ig subjc o h sicios ad Th udlig saml sac is a o b h collcio of all subss of siz of hich h a ad h a oucoms ha cosod o h v [ ] Accodig o h Poosiio - hav f aml 4-: A uchas of lcical comos bus hm i los of siz I is is olic o isc comos adoml fom a lo ad o acc h lo ol if all a o-dfciv If c of h los hav 4 dfciv comos ad 7 c hav ol ha ooio of los dos h uchas jc? Soluio: L A do h v ha h uchas accs a lo o 7 lo hasdfciv 4 dfcivs lo has A P A P A P Thom 4-: If HYP ~ h fo ach valu ad as ad ih a osiiv cosa lim

13 ahmaical Saisics 4 Poof: o gligibl facos gligibl facos gligibl facos Hc lim aml 4-: A lo of as is shid o a coma A samlig la dicas ha as a o b a a adom ad ihou lacm ad h lo accd if o mo o of hs as a dfciv H AC is usuall calld h accac umb Th oaig chaacisic cuv OC P h is h facio dfciv i h lo is all h sum of h h hgomic obabiliis f h Hov hav s ha h hgomic disibuio ca b aoimad b h biomial disibuio hich i u ca b aoimad b h Poisso disibuio h is lag ad is small This acl ou siuaio sic ad a isd i valu of i h ag o Thus OC P Thom 4-: If ~ HYP h

14 ahmaical Saisics 44 µ ad σ Poof: f W o ma h chag of vaiabls i h summaio ad lac ih i h domiao his bcom µ f ] [ I h summaio l ad i h domiao o ha Thus ] [

15 ahmaical Saisics 4 σ 46 Pobabili Gaig Fucios A imoa class of disc adom vaiabls is o i hich ss a cou ad cosul as ig valus: A ahmaical dvic usful i fidig h obabili disibuios ad oh ois of ig-valud adom vaiabls is h obabili gaig fucio Dfiiio 46-: L b a ig-valud adom vaiabl fo hich h i Th obabili gaig fucio P fo is dfid o b P i i P i i i Fo all valus of such ha P is fii Th aso fo callig P a obabili gaig fucio is ha h coffici of i P is h obabili Rad diffiaio of P ilds facoial moms fo h adom i vaiabl Dfiiio 46-: Th h facoial mom fo a adom vaiabl is dfid o b µ h is osiiv ig [ ] [ ] i o ha µ [] µ Th scod facoial mom µ [ ] [ ] as usful i fidig h vaiac fo biomial gomic ad Poisso adom vaiabls Thom 46-: If P is h obabili gaig fucio fo a ig-valud adom vaiabl h h h facoial mom of is giv b d P d P µ [ ] Poof: Sic P P i follos ha

16 ahmaical Saisics 46 ad i gal dp P 4P4 d d P P 4 P4 d d P P d Sig i ach of hs divaivs obai ad i gal P 4P4 µ [] P 4 P4 [ ] µ [ ] d P P [ ] µ [ ] d aml 46-: Fid h obabili gaig fucio fo a gomic adom vaiabl Th us P o fid h ma of a gomic adom vaiabl Soluio: o ha sic cao assum his valu Th P if < µ [] d P d Sic alad hav h mom gaig fucio o assis i fidig h moms of a adom vaiabl of ha valu is P? Th as is ha i ma b difficul o fid bu much asi o fid P Thus P ovids a addiioal ool fo fidig h moms of a adom vaiabl I ma o ma o usful i a giv siuaio Fidig h moms of a adom vaiabl is o h majo us of h obabili gaig fucio Is ima alicaio is i divig h obabili fucio fo oh ig-valud adom vaiabls