PROBABILITY AND STATISTICS FOR ENGINEERS

VŠB Techcal Uvery of Orava Faculy of Elecrcal Egeerg ad Comuer Scece Dearme of Aled Mahemac PROBABILITY AND STATISTICS FOR ENGINEERS Radm Brš Orava

PROBABILITY AND STATISTICS FOR ENGINEERS LESSON INSTRUCTIONS The lecure oe are dvded o chaer. Log chaer are logcally l o umbered ubchaer. Sudy Tme Emaed me o udy ad fully gra he ubjec of a chaer. The me aroxmae add hould oly be reaed a a gude. Learg Objecve Thee are am ha you eed o acheve a he ed of each chaer. They are baed o owledge ad ll requred. Exlaao Exlaao exad o he uded maeral. New erm are roduced ad exlaed more deal. Examle are gve. Summary Key dea are ummarzed cocluo of each chaer. If hey are o clear eough a h o, recommeded ha you go bac ad udy he chaer aga. Addoal Clue Examle ad Soluo Quz To mae ure ha you horoughly uderad he dcued ubjec, you are gog o be aed everal heorecal queo. You wll fd he awer he brace or a he ed of he exboo he SOLUTION KEYS eco.

Praccal Exerce A he ed of each log chaer raccal alcao of he heory reeed exerce. 3

EXPLORATORY DATA ANALYSIS Sudy Tme: 7 mue Learg Objecve Geeral Coce of Exloraory (Prelmary Sac Daa Varable Tye Sacal Characerc ad Grahcal Mehod of Preeg Qualave Varable Sacal Characerc ad Grahcal Mehod of Preeg Quaave Varable Exlaao Orgal goal of ac wa o collec daa abou oulao baed o oulao amle. By oulao we mea a grou of all exg comoe avalable for obervao durg acal reearch. For examle: If a acal reearch erformed abou hycal hgh of 5year old grl, he oulao wll be all grl currely aged 5. Coderg he fac ha he umber of oulao member uually hgh, he reearch wll be baed o he ocalled amle examao where oly ar of he oulao ued. The examed ar of he oulao called a amle. Wha' really mora o mae a defe eleco ha a rereeave of he whole grou a oble. There are everal way o acheve. To avod of omg ome eleme of he oulao he ocalled radom amle ued whch each eleme of oulao ha he ame chace of beg eleced. I goe whou ayg ha amle examao ca ever be a accurae a examg he whole oulao. Why do we do refer he?. To ave me ad mmze co (eecally for large oulao. 4

. To avod damagg amle derucve eg (ome e le examg choleerol blood ec., lead o he ermae damage of examed eleme. 3. Becaue he whole oulao o avalable. Now ha you ow ha ac ca decrbe he whole oulao baed o formao gahered from a oulao amle we wll move o o Exloraory Daa Aaly (EDA. Daa we oberve wll be called he varable ad her value varable vara. EDA ofe he fr e revealg formao hdde a large amou of varable ad her vara. Becaue he way of roceg varable deed mo o her ye, we wll ow exlore how varable are devded o dffere caegore. The varable dvo how he followg dagram. Varable Qualave (caegoral, lexcal... Quaave (umercal... geeral dvdg Dcree Couou Nomal Ordal Fe Deumerable dvdg baed o umber of vara vara Alerave Plural Qualave varable vara are exreed verbally ad hey l o wo geeral ubgrou accordg o wha relao bewee her value: Nomal varable ha equvale vara: moble o eher comare hem or or hem (for examle: ex, aoaly, ec. 5

Ordal varable form a rao bewee qualave ad quaave varable: dvdual vara ca be ored ad oble o comare oe aoher (for examle: cloh ze S, M, L, ad XL The ecod way of dvdg hem baed o umber of vara: Alerave varable ha oly wo oble oo (e.g. ex male or female, ec. Plural varable ha more ha wo oble oo (e.g. educao, ame, eye color, ec. Quaave varable exreed umercally ad ' dvded o: Dcree varable ha fe or deumerable umber of vara Dcree fe varable ha fe umber of vara (e.g. mah grade,,3,4,5 Dcree deumerable varable ha deumerable umber of vara (e.g. age (year, hegh (cm, wegh (g, ec. Couou varable ha ay value from  or from ome  ube (e.g. dace bewee ce, ec. Addoal clue Image ha you have a large acal grou ad you face a queo of how o be decrbe. Number rereeao of value are ued o relace he grou eleme ad hey become he bac arbue of he grou. Th wha we call acal characerc. I he ex chaer we are gog o lear how o e u acal characerc for varou ye of varable ad how o reree larger acal grou.. Sacal Characerc of Qualave Varable We ow ha a qualave varable ha wo bac ye omal ad ordal... Nomal Varable Nomal varable ha dffere bu equvale vara oe grou. The umber of hee vara uually low ad ha' why he fr acal characerc we ue o decrbe wll be frequecy. Frequecy (abolue frequecy defed a he umber of a vara occurrece of he qualave varable 6

I cae ha a qualave varable ha dffere vara (we decrbe her frequecy a, a acal grou (of value mu be rue ha: + +... + If you wa o exre he rooro of he vara frequecy o he oal umber of occurrece, we ue relave frequecy o decrbe he varable. Relave frequecy defed a: aleravely: [%] (We ue he ecod formula o exre he relave frequecy erceage o. For relave frequecy mu be rue ha: + + + K Whe qualave varable are roceed, good o arrage frequecy ad relave frequecy he ocalled frequecy able: FREQUENCY TABLE Value x Abolue frequecy Relave frequecy x x M M M x Toal The la characerc of omal varable he mode. Mode 7

defed a a vara ha occur mo frequely The mode reree a ycal eleme of he grou. Mode cao be deermed f here are more value wh maxmum frequecy he acal grou... Grahcal Mehod of Preeg reeg Qualave Varable The ac ofe ue grah for beer aaly of varable. There are wo ye of grah for aalyzg omal varable: Hogram (bar char Pe char Hogram a adard grah where vara of he varable are rereeed o oe ax ad varable frequece o he oher ex ex. Idvdual value of he frequecy are he he dlayed a bar (boxe, vecor, quared log, coe coe, ec. Examle: Clafcao Clafcao 8 8 6 6 4 4 8 8 6 6 4 4 3 4 Clafcao 4 Clafcao 8 8 6 6 4 4 8 8 6 6 4 4 3 4 Clafcao 8 6 4 8 6 4 8 3 3 4 3 4

Pe char reree relave frequece of dvdual vara of a varable. Frequece are reeed a rooro a ecor of a crcle. Whe we chage he agle of he crcle, we ca ge ellcal, hreedmeoal effec. Clafcao Clafcao 3 6 3 6 9 3 9 9 3 9 4 4 Clafcao Clafcao 3 6 3 6 9 9 3 4 9 9 3 4 REMEMBER! Decrbg he e char eceary. Marg dvdual ecor by relave frequece oly whou addg her abolue value o uffce. Examle: : A oo oll ha bee carred ou abou lauchg hgh chool fee. I reul are how o he followg char: 5% 5% YES NO Are he reul ereg? No maer how rue hey may be, recommeded ha he char be modfed a follow: 9

YES NO Wha he dfferece? From he ecod char obvou ha oly wo eole were aed he fr oe ad YES ad he ecod oe ad NO. Wha ca be leared from ha? Mae char uch a way ha her erreao aboluely clear. If you are reeed wh a e char whou abolue frequece mared o, you ca a yourelve wheher becaue of he auhor gorace or a delberae ba. Examle ad Soluo A obervaoal udy ha bee uderae o he ue of a ereco. The colleced daa are he able below. The daa made u of colour of car ha a hrough he ereco. Aalyze he daa ad erre he reul a grahcal form. red blue red Gree blue red red Whe gree gree blue Red Soluo: From he able obvou ha he colleced colour are qualave (lexcal varable, ad becaue here o order or comaro bewee hem, we ca ay hey are omal varable. For beer decro we creae a frequecy able ad we deerme he mode. We are gog o ree he colour of he ag vehcle by a hogram ad a e char. Color of ag car FREQUENCY TABLE Abolue frequecy Relave frequecy red 5 5/.4 blue 3 3/.5 whe /.8 gree 3 3/.5 Toal.

We oberved car oal. Mode red (.e. our amle mo car were red Colour of ag car Colour of ag car 6 5 4 3 3 5 red blue whe gree 3 red blue whe gree..3 Ordal Varable Now we are gog o have a loo a decrbg ordal varable. The ordal varable (ju le he omal varable ha varou verbal vara he grou bu hee vara ca be ored.e. we ca ell whch oe "maller" ad whch oe "bgger" For decrbg ordal varable we ue he ame acal characerc ad grah a for omal varable (frequecy, relave frequecy, mode vewed by hogram or e char lu wo oher characerc (cumulave frequecy ad cumulave relave frequecy hu cludg formao abou how hey are ored. Cumulave frequecy of he h vara m a umber of value of a varable howg he frequecy of vara le or equal he h vara E.g. we have a varable called "grade from Sac" ha ha he followg vara: "", "", "3" or "4" (where he be ad 4 he wor grade. The, for examle, he cumulave frequecy for vara "3", wll be equal umber of ude who ge grade "3" or beer. If vara are ored by her "ze" ( x < x < K < x he he followg mu be rue: m j j So elfevde ha cumulave frequecy h ("he hghe" vara equal o he varable. m

The ecod ecal characerc for ordal varable cumulave relave frequecy. Cumulave relave frequecy of h vara F a ar of he grou are he value wh he h ad lower vara. They are exreed by he followg formula: F j j Th ohg ele he relave exreo of he cumulave frequecy: F m Ju a he cae of omal varable we ca ree acal characerc ug frequecy able for ordal varable. I comaro o he frequecy able of omal varable alo coa value of cumulave ad cumulave relave frequece. FREQUENCY TABLE Value x Abolue frequecy Cumulave frequecy Relave frequecy Relave cumulave frequecy m F x m F x F + F + M M M M M x Toal m + F F +..4 Grahcal Preeao of Ordal Varable We brefly meoed hogram ad he e char a good way of reeg he ordal varable. Bu hee grah do' reflec vara org. To acheve ha, we eed o ue olygo (alo ow a Ogve ad Pareo grah. Frequecy Polygo a le char. The frequecy laced alog he vercal ax ad he dvdual vara of he varable are laced alog he horzoal ax (ored acedg order from he lowe" o he hghe". The value are aached o he le.

Frequecy olygo for he evaluao grade 8 6 4 frequecy 8 6 4 3 4 vara Ogve (Cumulave Frequecy Polygo a frequecy olygo of he cumulave frequecy or he relave cumulave frequecy. The vercal ax he cumulave frequecy or relave cumulave frequecy. The horzoal ax reree vara. The grah alway ar a zero, a he lowe vara, ad ed u a he oal frequecy (for a cumulave frequecy or. (for a relave cumulave frequecy. Ogve for he evaluao grade 4 35 cumulave frequecy 3 5 5 5 3 4 vara Pareo Grah a bar char for qualave varable wh he bar arraged by frequecy vara are o horzoal ax ad are ored from he hghe morace o he lowe 3

Noce he decle of cumulave frequecy. I dro a he frequecy of varable decreae. Examle ad Soluo Followg daa reree hr ze ha a cloh realer offer o ale: S, M, L, S, M, L, XL, XL, M, XL, XL, L, M, S, M, L, L, XL, XL, XL, L, M a Aalyze he daa ad erre reul a grahcal form. b Deerme wha erceage of eole bough hr of L ze maxmum. Soluo: a The varable qualave (lexcal ad hr ze ca be ored, herefore a ordal varable. For decro you ue frequecy able for he ordal varable ad you deerme he mode. Color of ag car FREQUENCY TABLE Abolue frequecy red 5 blue 3 whe gree 3 Relave frequecy Toal. Mode XL (he mo eole bough hr wh XL value For grahcal rereeao ue hogram, e grah ad cumulave frequecy olygo (you do' creae Pareo grah becaue moly ued for echcal daa. 4

Grahcal ouu: Hogram Sold hr Pe Char Sold hr 8 7 abolue frequecy 6 5 4 3 XL 3% L 7% S 4% M 7% S M L vara XL Cumulave Frequecy Polygo Sold hr 5 cumulave frequecy 5 5 S M L vara XL Toal ale were hr. b You ge he awer from he value of he relave cumulave frequecy for vara L. You ee ha 68% of eole bough hr of L ze ad maller.. Sacal Characerc of Quaave Varable To decrbe quaave varable, mo of he acal characerc for ordal varable decro ca be ued (frequecy, relave frequecy, cumulave frequecy ad cumulave relave frequecy. Aar from hoe, here are wo addoal oe: Meaure of locao hoe dcae a ycal drbuo of he varable value ad Meaure of varably hoe dcae a varably (varace of he value aroud her ycal oo 5

.. Meaure of Locao ad Varably The mo commo meaure of oo he varable mea. The mea reree average or ycal value of he amle oulao. The mo famou mea of quaave varable : Arhmecal mea x I defed by he followg formula: x! x where: x... are value of he varable... ze of he amle oulao (umber of he value of he varable Proere of he arhmecal mea:. ( x x! um of all dvero of varable value from her arhmecal mea equal o zero whch mea ha arhmecal mea comeae mae caued by radom error.. " ( a Î Â æ ç : ç x ç è! x Þ ( a + x! ö a + x ø f he ame umber added o all he value of he varable, he arhmecal mea creae by he ame umber 3. "( b Î Â æ ç : ç x ç ç è ö x ( bx!! Þ bx ø f all he varable value are mulled by he ame umber he arhmecal Mea creae accordgly Arhmecal mea o alway he be way o calculae he mea of he amle oulao. For examle, f we wor wh a varable rereeg relave chage (co dexe, ec. we ue he ocalled geomercal mea. To calculae mea whe he varable ha a form of a u, harmocal mea ofe ued. 6

Coderg ha he mea ue he whole varable value daa e, carre maxmum formao abou he amle oulao. O he oher had, ' very eve o he ocalled oulyg obervao (ouler. Ouler are value ha are ubaally dffere from he re of he value a grou ad hey ca dor he mea o uch a degree ha o loger reree he amle oulao. We are gog o have a cloer loo a he Ouler laer. Meaure of locao ha are le deede o he oulyg obervao are: Mode xˆ I he cae of mode we wll dffereae bewee dcree ad couou quaave varable. For dcree varable we defe mode a he mo freque value of he varable (mlarly a wh he qualave varable. Bu he cae of couou varable we h of he mode a he value aroud whch mo varable value are coceraed. For aeme of h value we ue horh. Shorh he hore erval wh a lea 5% of varable value. I cae of a amle a large a ( Î N (wh eve umber of value value le wh horh whch / (5% varable value. I he cae of a amle a large a + ( Î N (wh odd umber of value + value le wh hor whch abou / lu 5% varable value (/+/. The, he mode xˆ ca be defed a he cere of he horh. From wha ha bee ad o far clear ha he horh legh (o boudary boom boudary uque bu locao o. If he mode ca be deermed uambguouly we al abou umode varable. Whe a varable ha wo mode we call bmode. Whe here are wo or more mode a amle, uually dcae a heerogey of varable value. Th heerogey ca be removed by dvdg he amle o more ubamle (for examle bmode mar for ero' hegh ca be dvded by ex o wo umode mar wome' hegh ad me' hegh. Examle ad Soluo The followg daa how age of muca who erformed a a cocer. Age a couou varable. Calculae Mea, Shorh ad Mode for he varable. 8 7 43 9 47 4 34 34 4 35 Soluo: a Mea: 7

I h cae we ue arhmecal mea: x x + 8 + 7 + 43 + 9 + 47 + 4+ 34 + 34 + 4 + 35! 38.7 year The muca average age 38.7 year. b Shorh: Our amle oulao ha value. a odd umber. 5% of 5.5 ad he eare hgher aural umber 6 oherwe: /+/ /+/ / 6. Tha mea ha 6 value wll le he Shorh. Ad wha are he ex e? You eed o or he varable You deerme he ze of all he erval (havg 6 eleme where x < x < K < x + + 5 The hore of hee erval wll be he horh (ze of he erval x x +5 Orgal daa Sorg daa Sze of erval (havg 6 eleme 9 6 ( 35 9 8 9 ( 4 7 7 5 ( 4 7 43 34 9 ( 43 34 9 34 3 ( 47 34 47 35 47 ( 8 35 4 4 34 4 34 43 4 47 35 8 From he able you ca ee ha he hore erval ha he value of 9. There oly oe erval ha correod o h ze ad ha : 34 ; 43. Shorh 34 ; 43 ad ha mea ha half of he muca are bewee 34 ad 43 year of age. c Mode: Mode defed a he ceer of horh: 34 + 43 x ˆ 38.5 8

Mode 38.5 year whch mea ha he ycal age of he muca who erformed a he cocer wa 38.5 year. Amog oher characerc decrbg quaave varable are quale. Thoe are ued for more dealed llurao of he drbuo of he varable value wh he coe of he oulao. Quale Quale decrbe locao of dvdual value (wh he varable coe ad are rea o oulyg obervao mlarly le he mode. Geerally he quale defed a a value ha dvde he amle o wo ar. The fr oe coa value ha are maller ha gve quale ad he ecod oe wh value larger or equal ha he gve quale. The daa mu be ored acedgly from he lowe o he hghe value. Quale of varable x ha earae % maller value from he re of he amle (.e. from (% value wll be called % quale ad mared x. I real lfe you mo ofe come acro he followg quale: Quarle I cae of he fourar dvo he value of he varae correodg o 5%, 5%, ad 75% of he oal drbuo are called quarle. Lower quarle x,5 5% quale dvde a amle of daa a way ha 5% of he value are maller ha he quarle,.e. 75% are bgger (or equal Meda x,5 5% quale dvde a amle of daa a way ha 5% of he value are maller ha he meda ad 5% of value are bgger (or equal Uer quarle x,75 75% quale dvde a amle of daa a way ha 75% of value are maller ha he quarle,.e. 5% are bgger (or equal Examle: Daa 6 47 49 5 43 4 7 39 43 4 36 Daa acedg order 6 7 5 36 39 4 4 43 43 47 49 Meda 4 Uer quarle 43 Lower quarle 5 The dfferece bewee he ad 3rd quarle called he IerQuarle Rage (IQR. 9

IQR x x.75.5 Examle: Daa 3 4 5 6 6 6 7 7 8 9 Uer quarle 7 Lower quarle 4 IQR 7 4 3 Decle x.; x.;... ; x.9 The decle dvde he daa o equal rego. Percele x. ; x. ; ; x.99 The ercele dvde he daa o equal rego. For examle, he 8 h ercele he umber ha ha 8% of value below ad % above. Raher ha coug 8% from he boom, cou % from he o. Noe: The 5 h ercele he meda. Mmum x m ad Maxmum x max x x m,.e. % of value are le ha mmum x,.e. % of value are le ha maxmum max x There he followg roce o deerme quale:. The amle oulao eed o be ordered by ze. The dvdual value are equeced o ha he malle value a he fr lace ad he hghe value a h lace ( he oal umber of value 3. % quale equal o a varable value wh he equece z where: z +.5 z ha o be rouded o eger!!! REMEMBER!!! I cae of a daa e wh a eve umber of value he meda o uquely defed. Ay umber bewee wo mddle value (cludg hee value ca be acceed a he meda. Mo ofe he mddle value. We are ow gog o dcu he relao bewee quale ad he cumulave relave frequecy. The value deoe cumulave relave frequecy of quale x

.e. relave frequecy of hoe varable value ha are maller ha quale x. Quale ad cumulave relave frequecy are vere coce. Grahcal or abular rereeao of he ordered varable ad arorae cumulave frequece ow a drbuo fuco of he cumulave frequecy or emrcal drbuo fuco. Emrcal Drbuo Fuco F(x for he Quaave Varable We u he amle oulao acedg order (x <x < <x ad we deoe (x a relave frequecy of he value x. For emrcal drbuo fuco F(x mu he be rue ha: F ì for x x ï j í for x j < x x j +, ï î for x < x ( x ( x j The emrcal drbuo fuco a moooou, creag fuco ad ru from he lef. ( x lm F( x F( x F(x x x + (x (x... x x x 3 x x x MAD MAD a hor for Meda Abolue Devao from he meda. MAD deermed a follow:. Order he amle oulao by ze. Deerme he meda of he amle oulao

3. For each value deerme abolue value of devao from he meda 4. Pu abolue devao from he meda acedg order by ze 5. Deerme he meda of he abolue devao from he meda.e. MAD Examle ad Soluo There he followg daa e:, 8, 7, 43, 9, 47, 4, 34, 34, 4, 35 (he daa from he revou examle. Deerme: a All quarle b IerQuarle Rage c MAD d Draw he Emrcal Drbuo Fuco Soluo: a You eed o deerme Lower Quarle x,5 ; Meda x,5 ad Uer Quarle x,75. Fr, you order he daa by ze ad ag a equece umber o each value. Orgal daa Ordered daa Sequece 9 8 7 7 3 43 34 4 9 34 5 47 35 6 4 4 7 34 4 8 34 43 9 4 47 35 8 Now you ca dvde he daa e o quarle ad mar her varable value accordgly: Lower Quarle x,5 :.5; Þ z x.5 +.5 3.5 3 Þ x.5 7.e. 5% of muca are uder 7 (75% of hem are 7 year old or older. Meda x,5 :.5; Þ z x.5 +.5 6 Þ x.5 35.e. a half of he muca are uder 35 (5% of hem are 35 year old or older. Uer Quarle x,75 :.75; Þ z x.75 +.5 8.75 9 Þ x.75 43.e. 75% muca are uder 43 (5% of hem are 43 year old or older. b IerQuarle Rage IQR:

IQR x.75 x.5 43 7 6 c MAD If you wa o deerme h characerc you mu follow defo (he meda of abolue devao from he meda. x.5 35 Orgal daa x Ordered daa y Abolue value of devao of he ordered daa from her meda y x.5 Ordered abolue value M 9 6 9 35 8 3 35 7 7 8 7 35 43 34 34 35 6 9 34 34 35 7 47 35 35 35 8 4 4 6 4 35 8 34 4 7 4 35 34 43 8 43 35 3 4 47 47 35 6 35 8 47 8 35 47 MAD M.5.5; Þ z x.5 +.5 6 Þ x.5 8 (MAD a meda abolue devao from he meda.e. 6 h value of ordered abolue devao from he meda MAD 8. d The la a wa o draw he Emrcal Drbuo Fuco. Here defo: F ì forx x ï j í forx j < x x j+, ï ïî forx < x ( x ( x j Arrage he varable value a well a her frequece ad relave frequece acedg order ad wre hem dow he able. The derve he emrcal drbuo fuco from hem: 3

Orgal daa x Ordered daa a Abolue frequece of he ordered value Relave frequece of he ordered value Emrcal drbuo fuco F(a 9 / 8 / / 7 7 / / 43 34 / 3/ 9 35 / 5/ 47 4 / 6/ 4 4 / 7/ 34 43 / 8/ 34 47 / 9/ 4 8 / / 35 A by defo he emrcal drbuo fuco F(x equal for each x <9; F(x equal / for all x>9; F(x equal / + / for all 7 x> ; ad o goe o. X ( ; 9 ( 9 ; ( ; 7 ( 7 ; 34 ( 34 ; 35 F(x / / 3/ 5/ X ( 35 ; 4 ( 4 ; 4 ( 4 ; 43 ( 43 ; 47 ( ; 8 8 ; F(x 6/ 7/ 8/ 9/ / / 47 ( Mea, mode ad meda (.e. meaure of locao reree magary cere of he varable. However, we are alo ereed he drbuo of he dvdual value of he varable aroud he cere (.e. meaure of varably. The followg hree acal characerc allow decro of he amle oulao varably. Shorh ad IerQuarle Rage are clafed a meaure of varably. Samle Varace Emrcal drbuo fuco,,,8 F(x,6,4,, 4 6 8 x he mo commo meaure of varably 4

5 The amle varace gve by: ( x x Samle Varace he um of all quared devao from her mea dvded by oe le ha he amle ze Geeral roere of he amle varace are for examle: The amle varace of a coa umber zero I oher word: f all varable value are he ame, he amlg ha zero dffuee ( ( ( : y y x a y x x a Þ ú ú ú û ù ê ê ê ë é + Ù ø ö ç ç ç ç è æ ÎÂ " I oher word: f you add he ame coa umber o all varable value, he amle varace doe chage ( ( ( : b y y bx y x x b Þ ú ú ú û ù ê ê ê ë é Ù ø ö ç ç ç ç è æ Î Â " I oher word: f you mully all varable value by a arbrary coa umber (b he amle varace creae by quare of h coa umber (b Dadvaage of ug he amle varace a a meaure of varablly ha emloy quared value of he varable. For examle: f he varable reree cah deomaed EUR, he he amle varao of h varable wll be EUR. Tha why we ue aoher meaure of varably called adard devao. Sadard Devao calculaed by he quare roo of he varace

Aoher dadvaage of ug he amle varao ad he adard devao ha varably of he varable ca be comared dffere u. Whch varable ha bgger varably hegh or wegh of a adul? To awer ha, Coeffce of Varao ha o be ued. Coeffce of Varao V x reree relave meaure of varably of he varable x ad ofe exreed a a erceage he rao of he amle adard devao o he amle mea: V x x Examle ad Soluo A able gla maufacurer ha develoed le exeve echology for mrovg he frerea gla. gla able hee were eleced for eg. Half of hem were reaed by he ew echology whle he oher half wa ued for comaro. Boh lo were eed by fre ul hey craced. Thee are he reul: Crcal emeraure (gla craced [ o C] Old echology x New echology y 475 485 436 39 495 5 483 46 46 488 Comare boh echologe by mea of bac characerc of he exloraory aaly (mea, varao, ec.. Soluo: Fr you comare boh echologe by he mea: Mea for he old echology: Mea for he ew echology: 6

Baed o he calculaed mea he ew echology could be recommeded becaue he emeraure ca whad 6 o C hgher. ow you deerme he meaure of varably The old echology: Samle Varace: Sadard Devao: New echology: Samle varace: Sadard devao: Samle varace (adard devao for he ew echology gfcaly larger. Wha he oble reao? Loo a he grahcal Crcal emeraure rereeao of he colleced daa. Crcal emeraure are much more read ou whch 6 mea h echology o fully uder corol ad ue ca' guaraee hgher roduco qualy. I h cae he crcal emeraure ca eher be much hgher or much lower. For ha reao recommeded ha hould be ubjeced o addoal reearch. Thee cocluo 3 are baed oly o exloraory aaly. Sac Old New rovde u wh more exac mehod for aalyg Techologe mlar roblem (hyohe eg. Temeraure 7

Now we are gog o reur o exloraory aaly a uch. We meoed ouler. So far we ow ha ouler are varable value ha are ubeally dffere from he re of he value ad h mac o mea. How ca hee value be defed? Idefcao of he Ouler I he acal racce we are gog o come acro a few mehod ha are caable of defyg ouler. We'll meo hree ad go hrough hem oe by oe.. The ouler ca be every value x ha by far exceed,5 IQR lower (or uer quale. [( x < x.5iqr Ú ( x < x +. 5IQR ] Þx aouler.5. 75. The ouler ca be every value x of whch he abolue value of he zcore greaer he 3. x x z core ( z core. > 3 Þx aouler 3. The ouler ca be every value x of whch he abolue value of he medacore greaer he 3. meda core. x x.5.483. MAD ( meda core. > 3 Þ x a ouler Ay of he hree rule ca be ued o defy ouler reallfe roblem. The Z ax "le rc" ha he medaax o ouler. I' becaue eablhg he z ax baed o mea ad adard devao ad hey are rogly flueced by oulyg value. Meawhle, eablhg he medaax baed o meda ad MAD ad hey are mmue o ouler. Whe you defy a value a a ouler you eed o decde ye ule caued by: mae, yg error, huma error, echology whm, ec. faul, reul of wrog meaureme, ec. If you ow he ouler caue ad mae ure ha wll o occur aga, ca be cleared from he roce. I oher cae you mu coder carefully f by geg rd of a ouler you wo loe mora formao abou eve wh low frequece. 8

The oher characerc decrbg qualave varable are ewe ad uro. Ther formula are raher comlex herefore ecalzed ofware ued for he calculao. Sewe Sewe defed a aymmery he drbuo of he varable value. Value o oe de of he drbuo ed o be furher away from he "mddle" ha value o he oher de. The followg formula ued: Sewe erreao: a... varable value are drbued ymmercally aroud he mea a >... value maller ha he mea are redoma a <... value larger ha he mea are redoma 6 6 6 5 5 5 4 4 4 3 3 3 3 4 5 6 7 3 4 5 6 7 3 4 5 6 7 α α> α< Kuro Kuro reree cocerao of varable value aroud her mea. The followg formula ued o ge value: b ( + ( ( ( 3 ( x x 4 4 3 ( ( ( 3 Kuro erreao: 9

b... Kuro correod o ormal drbuo b >... "eaed" drbuo of he varable b <... "fla" drbuo of he varable 7 3 6 5 4 3 3 4 5 6 7 8 6 4 3 4 5 6 7 5 5 5 3 4 5 6 7 β β> β< We have ow defed all umercal characerc of he quaave varable Nex we are gog o ae a loo a how hey ca be erreed grahcally... Grahcal Mehod of Preeg Quaave Varable Box lo A box lo a way of ummarzg a daa e o a erval cale. I ofe ued exloraory daa aaly. I a grah ha how he hae of he drbuo, cere o, ad varably. The reulg cure co of he mo exreme value he daa e (maxmum ad mmum, he lower ad uer quarle, ad he meda. A box lo eecally helful for dcag wheher a drbuo ewed ad wheher here are ay uuual obervao (ouler he daa e. horh 6 5 4 3 Ouler Max Uer Quarle Meda Lower Quarle M Noce: A box lo coruco beg by marg ouler ad he oher characerc (m, max, quarle ad horh. 3

Sem ad Leaf Plo A we aw, mlcy a advaage of he box lo. However, formao abou ecfc value of he varable mg. The mg umerc value would have o be ecfcally mared dow oo he grah. The Sem ad leaf lo wll mae u for ha lmao. We have a varable rereeg average moh alary of ba emloyee he Czech Reublc. Average moh ay [CZK],654 9,765 8,675,435 9,675,343 8,786 5,4 8,675 7,3 6,73 6,878 5,657 9,754 9,543 9,435,647,453 9,987,34 Average moh ay [CZK] daa acedg order 6,73 6,878 7,3 8,675 8,675 9,435 9,543 9,675 9,754 9,765 9,987,34,343,647,654,435,453 5,4 5,657 8,786 How do we brg he daa oo he grah? The lace value ha are regarded a umora are gored ad daa o he hgher lace are u order. We are eecally ereed he value from he hrd (hudred lace. The value o he fourh (houad lace are wre dow acedg order hu creag a em. Uder he grah we aed a em wdh ha wll alo ac a a coeffce ued o mully value he grah. The ecod colum of he grah ow a leave are he umber rereeg "mora" lace value. They are wre dow correodg row. The hrd colum abolue frequecy for arcular row. For examle: he fr row he grah reree wo value (6.7 ad 6.8* 3 CZK.e. 6,7 CZK ad 6,8 CZK, he xh row reree wo value oo (.4 ad.4* 3 CZK,.e. wo emloyee have he average moh ay of,4 CZK, ec. Sem 6 78 7 8 66 9 456779 6 3366 4 44 5 46 8 7 * 3 Sem wdh Leave Frequece There are varou modfcao of h grah. For examle he hrd colum could ore cumulave frequece ad he meda row he abolue frequecy how arehee. From h row he abolue frequece eher cumulae from he malle value or dmh from he hghe value a ee o he cure. Sem * 3 6 78 7 3 8 66 5 9 456779 (6 3366 9 44 5 5 46 3 8 7 Sem wdh Leave Cumulave frequece 3

Fally, you eed o ee md ha here are dffere way of corucg a em ad leaf lo ad you eed o be aware of oe arcular roblem. Nowhere ad whch lace value of he varable are mora ad whch oe are o. Th lef o he oberver. However here a o follow. A log em wh hor leave ad a hor em wh log leave dcae correc choce of cale. Loo a he cure. 66788999999 558 9 * 4 Quz. Wha exloraory ac cocered wh?. Characerze he bac ye of varable. 3. Whch acal characerc ca be coaed frequecy able (for wha ye of varable? 4. Wha are he ouler ad how do you defe hem? 5. Whch characerc are eve o ouler? a Meda b Arhmecal Mea c Uer Quarle 6. How do you dec he qualave (quaave varable? 7. The followg box lo reree ude earg durg holday. Mar aeme ha do o correod o he dlayed realy: a A ude eared 9 houad CZK maxmum. b Ierquarle rage aroxmaely houad CZK. c Half of he ude eared le ha houad CZK. d Shorh rougly a erval of (5;5 houad CZK 3

Praccal Exerce Exerce : The followg daa reree car maufacurer coure of org. Aalyze he daa (frequecy, relave frequecy, cumulave frequecy ad cumulave relave frequecy, mode ad erre he grahcal form (hogram, e char. USA USA Germay Czech Re. Germay Germay Germay Czech Re. Czech Re. Czech Re. USA Germay Exerce : The followg daa reree cuomer wag me (m whe dealg wh he cuomer ervce. Draw box lo ad em ad leaf lo. 8 9 5 5 4 3 7 Exerce 3: A raffc urvey wa carred ou o eablh a vehcle cou a a ereco. A ude daa collecor recorded he umber of car wag queue each me he gree lgh jumed o. Thee are h/her oucome: 3 5 3 3 5 7 8 8 6 8 5 5 8 5 4 7 5 6 3 4 8 4 4 5 5 4 3 3 4 9 6 5 3 5 3 5 7 5 8 4 4 3 5 6 4 6 9 3 6 3 5 3 5 3 7 6 3 7 5 6 Draw box lo, emrcal drbuo fuco ad calculae he mea, adard devao, horh, mode ad erquarle rage. 33

. PROBABILITY THEORY Sudy Tme: 7 mue Learg Objecve you wll be able o Characerze robably heory Exla geeral oo of robably heory Exla ad ue geeral relao bewee eve Exla a oo of robably Defe robably by bac axom Defe roere of robably fuco Ue a codoal robably Exla heorem of oal robably ad Baye heorem 34

.. Iroduco o Bac Coce Sudy me: mue Learg Objecve Characerze robably heory Exla geeral oo of robably heory Exlaao Probably heory he deducve ar of ac. I uroe o gve a rece mahemacal defo or rucure o wha ha o far bee a uve coce of radome. Defg radome wll allow u o mae exac robably aeme. For examle whe dcug aocao, we could oly mae rough aeme erm of edece. Mahemacally, robably a e fuco whch mea remed o e. Therefore, we are begg h dcuo by exlag he fudameal aure of e ad he bac oerao erformed o e ad eleme a he ma dea behd he robably fuco. Geeral Noo of he Probably Theory Defo of a Se e A a colleco of eleme. Eleme are bac uve mahemacally udefed ee. To defe a e, eceary o be able o deerme wheher ay eleme cluded or o cluded he e. The oo of cluo alo a uve udefed coce. Defo of Elemeary Eve I robably heory, he robably aumo are baed o eleme ha are ar of a e. The e eleme are called elemeary eve. I racce, hee elemeary eve may be meaurable by u, cae, amle, o, ec. Examle: {head or al} whe og a co {,,3,4,5,6} whe hrowg a dce The e of all reul wll be deoed by Ω ad called amle ace (of he elemeary eve. The elemeary eve {ω} a ube of he Ω e whch coa oe eleme ω from Ω e, ωî Ω. The he eve A wll be a arbrary ube of Ω, A Ì Ω. 35

From acal daa we ca ealy eablh ha hare of boy bor arcular year wh reec o all bor chldre movg aroud 5.5%. Dee he fac ha dvdual cae we ca' redc ex of a chld we ca mae a relavely accurae gue abou how may boy here are amog chldre. A he examle ugge, relave frequece of ome eve are ablzed wh creaed reeo of cera value. We hall call h heomeo Sably of he Relave Frequece ad a emrcal fudameal rcle of he robably heory. Relave frequecy umber (A/ where a oal umber of radom obervao ad (A a umber of obervao wh a A reul. Summary Probably heory a mahemacal brach ug axomac logcal rucure. Mahemacal ac a cece cocered wh daa mg, daa aaly, ad formulag reul Radom obervao every fe roce where he reul o e by codo uder whch ru. Samle ace Ω a e of all oble oucome of a a radom obervao. Relave frequece of ome eve wh creaed reeo dcae ome level of ably. 36

.. Oerao wh he elemeary eve Sudy me: mue Learg Objecve Tye of elemeary eve Geeral relao bewee eve Exlaao Wha are he ye of elemeary eve? If a elemeary eve ω Î Ω (ω Î A occur he you ca ay ha a eve A ha occured. Th reul deoed by ω Î A ad favorable o he eve A. Cera eve he eve whch occur wh each radom exerme. I equvale o he Ω e. The cera eve for examle occur whe you hrow a dce ad you ed u wh oe of he x umber:,,3,4,5,6 Imoble eve he eve whch ever occur a exerme. I wll be deoed byæ. The moble eve for examle would be hrowg umber 8 (wh he ame dce. Wha are relao bewee eve? Oerao o Se The oerao of uo, ereco, comlemeao (egao, ubraco, he coce of ube, ad he ull e ad uveral e or amle ace are he algebra of e. Iereco A Ç B I he e of all eleme ha are boh A ad B. Grahcal examle: AI B w w Î A Ù w Î B { } 37

A B Examle for hrowg a dce: Number, 3 or 4 are hrow a eve A ad a eve umber hrow a eve B. I obvou ha A Ç B {,4}. Uo A È B I he e of all eleme ha are eher A or B. Grahcal examle: AU B w w Î A Ú w Î B { } A B Examle hrowg a dce: Eve A {,3,4} ad eve B a eve umber. I obvou ha A È B {,,3,4,6}. Djo eve A Ç B Æ Two eve A ad B ca occur ogeher. They have o commo reul. Examle for hrowg a dce: You hrow a eve umber a eve A ad a odd umber a eve B. Thee eve ever have he ame reul. If eve A occur he eve B ca hae. Sube (Subeve A Ì B A a ube of B f each eleme of A alo a eleme of B. I mea ha f eve A occur ha eve B occur a well. Grahcal examle: A Ì B Û { w ÎA Þ w Î B} 38

B A Examle for hrowg a dce: You hrow umber a eve A ad you hrow a eve umber a eve B. The eve A ubeve of eve B. Eve A ad B are equvale A B f A Ì B ad a he ame me B Ì A. Examle for hrowg a dce: You hrow a eve umber a eve A ad you hrow a umber ha dvdable by umber a eve B. Thee eve are equvale. Subraco AB The e of all eleme ha are A bu o B A B A I B Grahcal examle: A B { w w Î A Ù w Ï B} A B Examle for hrowg a dce: You hrow a umber greaer ha a eve A ad you hrow a eve umber a eve B. Subraco of he wo eve A B {3,5}. Comleme of eve A (ooe eve The e of all eleme ha are o A. A w w Ï A { } 39

Grahcal examle: A W Examle for hrowg a dce: You hrow a eve umber a eve A ad you hrow a odd umber a eve A DeMorga' Law DeMorga' Law are logcal cocluo of he fudameal coce ad bac oerao of he e heory. Law o. The e of all eleme ha are eher A or B. AU B AI B A B Law o. The e of all eleme ha are eher o A or o B (ha are o he ereco of A ad B. AI B AU B 4

Muually djo e ad arog he amle ace The colleco of e {A, A, A 3,...} aro he amle ace W: AI A j Æ for ¹ j W U A A A A A 5 A A 6 7 W 4 A 3 4

.3. Probably Theory Sudy me: 3 mue Learg Objecve Noo of robably Bac heorem ad axom of robably Tye of robably Codoal robably Theorem of oal robably ad Baye' heorem Exlaao Coce of Probably, Clacal Defo Le u coder a exerme wh N oble elemeary, muually excluve ad equally robable oucome A, A,,... A N. We are ereed he eve A whch occur f ayoe of M elemeary oucome occur, A, A,,... A M,.e. A M U A Sce he eve are muually excluve ad equally robable, we roduce he robably of he eve A, P(A a follow: P(A where umber of oucome of ere oal umber of oble oucome Th reul very mora becaue allow comug he robably wh he mehod of combaoral calculu; alcably however lmed o he cae whch he eve of ere ca be decomoed a fe umber of muually excluve 4

ad equally robable oucome. Furhermore, he clacal defo of robably eal he obly of erformg reeaed ral; requre ha he umber of oucome be fe ad ha hey be equally robable,.e. defe robably reorg o a coce of frequecy. We are alo gog o roduce Axomac Probably Defo. Axomac Probably Defo Probably ace a rad (Ω, S, P where ( Ω amle ace (eleme of Ω are elemeary eve ( S a e of ube of Ω where he followg rue: a ΩÎS; b f AÎS he A Ω A Î S; c f A, A, A 3,... Î S he U Eleme of S are called eve. A Î S ( P a fuco derved from S where he followg rue: a P(Ω Probable are caled o le he erval [,]; b P( A P(A for every AÎS ; c For a colleco of muually djo e, he robably of her uo equal o he um of her robable. If A Ç B Æ, he P{A U B} P{A} + P{B} I geeral, A Ç A P{ U j A } Æ, ", j ; ¹ j, P{ A } Fuco P called robably meaure or mly robably. Examle for hrowg a dce: Ω {,,3,4,5,6,}, S a e of ube of Ω (omeme we deoe S by ex Ω ad robably defed by carda P(A where carda umber or A eleme he e. 6 Bac Theorem of Probably The followg heorem are logcal cocluo of he hree bac robably axom oulaed o far.. For djo eve A ad B he followg rue: A Ç B Æ he 43

P{A U B} P{A} + P{B}. If for wo eve A,B: B Ì A he P{ B} P{ A} Noe ha A aroed by B ad comleme, ad hece P{A} um of hee wo ar 3. For every eve A he followg rue: P { A} P{A} The uo of he wo e he amle ace, he ereco he ull e. 4. I hold ha: P { Æ } 5. I hold ha: P{ B A} P{B} P{BI A} Noe ha BA ad B ereco A are wo djo e whoe uo B 6. Parcularly f A Ì B he P{ B A} P{B} P{A} 7. For arbrary eve A,B hold ha: P{A È B} P{A} + P{B} P{A Ç B} 8. I accordace wh he de Morga' law: P{ A È B} P{ A È B} P{ A Ç B} Defo of Codoal Probably The defo of codoal robably deerme how robable adju o chagg codo. Whe we ay ha he codo B ale, we mea ha he e B ow o have occurred ad herefore he re of he amle ace he comleme of B ha zero robably. Uder hee ew crcumace, he reved robably of ay oher eve, A, ca be deermed from he followg defo of codoal robably: P{A B} P{A Ç B} P{B} By h formula, he robably of ha ar of he eve A whch B or erec wh B reved uward o reflec he codo ha B ha occurred ad become he ew robably of A. I aumed ha he robably of B o zero. 44

P{A B} robably of he eve A codoed by he eve B Codoal Probably Defo of Ideedece If he codo ha B ha occurred doe o affec he robably of A, he we ay ha A deede of B. P {A B} P{A} From he defo of codoal robably, h mle P {A} P{AI B} P{B} ad hece, P{A I B} P{A} P{B} I clear from h demorao ha f A deede of B, he B alo deede of A. Examle for hrowg a dce: If for eve A you hrow he fr hrow, ad for eve B you hrow he ecod hrow, ad for eve C A Ç B you hrow boh hrow, he he followg rue: P{C} P{A Ç B} P{A} x P{B} 6 6 36 Theorem of Toal Probably If a colleco of e {B, B, B 3,..., B } aro he amle ace W, ha, B I B U B j Æ ; " W ¹ j 45

he for ay e A (P{A} he amle ace W, P{A} P{A B } P{B } 7 B B B 3 B B 5 B B 6 7 4 Ω Proof: Sce he colleco of e {B, B, B 3,..., B } aro he amle ace W, P{A} P{AI B } From he defo of codoal robably P{AI B } P{A B }P{B } Baye' Theorem If he colleco of e {B, B, B 3,..., B } aro he amle ace W, he P{B A} P{A B } P{B P{A B }P{B } } Proof: From he defo of codoal robably, P{B A} P{B I A} P{A} P{A}P{B } P{A} The roof come from ubug P{A} a defed by he Theorem of Toal Probably. Grahcal rereeao of Baye' heorem (he mared area reree eve A: 46

47

PROBABILITY THEORY EXAMPLES AND SOLUTIONS Examle ad Soluo Probably of he exguhg yem falure %. Probably of he alarm yem falure % ad robably ha boh yem fal 4%. Wha he robably ha: a a lea oe of he yem wll ay worg? b boh yem wll ay worg? Soluo: H... exguhg yem wor S... alarmg yem wor I ow ha: P ( H, P ( S, P ( H Ç S, 4 You mu fd: ada P( H È S There are wo oble oluo: By defo: Eve H ad S are o djo eve ad hece: P ( H S P( H + P( S P( H Ç S È, bu would be a roblem o deerme P( H Ç S By he ooe eve from he de Morga law you ca ay: P P ( H È S P( H È S P( H Ç S, ( H È S,4, 96 The robably (ha a lea oe yem wll be worg 96%. adb P( H Ç S We ca olve by he defo: ( P ( H S P( H S P( S P( S H P( H Ç, 48

becaue here lle formao abou how deede he falure are o dvdual yem. Hece we ry o ue he ooe eve: P P ( H Ç S P( H Ç S P( H È S [ P( H + P( S P( H Ç S ], ( H Ç S [ P( H + P( S P( H Ç S ] [, +,,4], 74 The robably (ha boh yem wll be worg 74%. Examle ad Soluo ude aed mahemac ad hyc exam. 3 of hem faled boh exam. 8 of hem faled oly he mah exam ad 5 of hem faled o a oly he hyc exam. Wha he robably ha a radom ude: a aed he mah exam f you ow ha he had faled he hyc exam b aed he hyc exam f you ow ha he had faled he mah exam c aed he mah exam f you ow ha he had aed he hyc exam Soluo: M... he aed he mah exam F... he aed he hyc exam You ow ha: You mu fd: P P P ( M Ç F ( M Ç F ( M Ç F ada P ( M F 3 8 5 by he defo of codoal robably: P P ( M F ( M F P P ( M Ç F P( F P P( M Ç F ( M Ç F + P( M Ç F P( M Ç F ( M Ç F + P( M Ç F 5, 5 3 + 5 35 @,4 7 49

The robably (ha he aed he mah exam f you ow ha he had faled he hyc exam 4%. adb P ( F M he ame way a ada: P ( F M ( P P F M P ( F Ç M P( M P P( F Ç M ( F Ç M + P( F Ç M P( F Ç M ( F Ç M + P( F Ç M 8 8 3 + 8 38, 4 @. 9 The robably (ha he aed he hyc exam f you ow ha he had faled he mah exam %. adc P ( M F from he defo: P ( M F ( M Ç F, P P(F here are wo oble: P( M F P ( M Ç F P( M Ç F P( F P( F P( M È F [ P( F Ç M + P( F Ç M ] [ P( F Ç M + P( F Ç M ] + [ P( F Ç M + P( F Ç M ] P( F Ç M ] [ P( F Ç M + P( F Ç M ] [ P( F Ç M + P( F Ç M + P( F Ç M ] [ P( F Ç M + P( F Ç M ] You record he daa o a able: é 5 8 3 ù ê + + ú ë û é 5 3 ù ê + ú ë û [ P( F + P( M P( F Ç M ] [ P( F Ç M + P( F Ç M ] 77 85 77 @.9 85 They aed he mah They faled he mah Toal exam exam They aed he hyc exam 8 They faled he hyc exam 5 3 35 Toal 38 ad you calculae ad fll he remag daa: 5

How may ude aed he hyc exam? I he oal umber of ude ( mu he umber of ude who faled he hyc exam (35 ad ha 85. Aalogouly for he umber of ude who aed he mah exam: 38 8. Ad for he umber of ude who aed boh exam: 8 5 77. They aed mah exam They faled o a mah exam They aed hyc exam 77 8 85 They faled o a hyc exam 5 3 35 Toal 8 38 The robable are: 77 85 P ( M Ç F ; P( F, whch mle ha: ( M Ç F P P( M F P( F 77 85 77 @,9 85 Toal The robably (ha he aed he mah exam f you ow ha he had aed he hyc exam 9%. Examle ad Soluo (Alcao of Baye' Theorem I a famou elevo how, he wer of he relmary roud gve he ooruy o creae he wg. The coea reeed wh hree cloed door ad old ha behd oe of he door here a ew car whle behd he oher wo door here are goa. If he coea correcly elec he door o he car he or he wll w he car. The ho a he coea o mae a eleco ad he oe oe of he oher wo door o ee f here a goa. The coea he gve he oo of wchg h/her choce o he oher door ha ll rema cloed. Should he or he go for ha choce? 3 5

Soluo: The amle ace co of hree oble arrageme {AGG, GAG, GGA}. Aume ha each of he hree arrageme ha he followg robable: P{AGG} P{GAG} 3 P{GGA} where + + 3. Aume ha he coea' fr choce Door # ad he ho oe Door #3 o reveal a goa. Baed o h formao we mu reve our robably aeme. I clear ha he ho cao oe Door #3 f he car behg. P{Door #3 GGA} Alo, he ho mu oe Door #3 f Door # lead o he car ce he cao oe Door #, he coea' choce. P{Door #3 GAG} Fally f he car behd he coea' fr choce, Door #, he ho ca chooe o oe eher Door # or Door #3. Suoe he chooe o oe Door #3 wh ome robably q. P{Door #3 AGG} q The accordg o Baye' Theorem, we ca comue he reved robably ha he car behd Door # a follow: P{GAG P{Door #3 GAG} P{GAG} Door #3} P{Door #3} Subug he ow value o h equao we oba, P{GAG Door #3} ( q + ( + ( 3 q + Thu he robably ha he car behd Door # afer he ho ha oeed Door #3 greaer ha 5% f, q <. I h cae, he coea hould mae aoher choce. Uder ormal crcumace, where he orgal robable of he hree arrageme,, are equal, ad he ho chooe radomly bewee Door # ad Door #3, he reved robably of Door # leadg o he car wll be greaer 5%. Therefore, ule he coea ha a rog a ror belef ha Door # 5

coceal he car, ad/or beleve ha he ho wll refer o oe Door #3 before Door #, he hould wch h/her choce. A he above dagram llurae, f he orgal robable of all hree arrageme are equal ad he ho chooe radomly whch door o oe, he of he oe half of he amle ace covered by oeg Door #3, wo hrd fall he rego occued by arrageme GAG. Therefore, f he ho oe Door #3, Door # become wce a lely a Door # o coceal he auomoble. Summary Radom exerme every fe roce whoe reul o deermed advace by codo ru uder ad whch a lea heorecally fely reeaable. Poble reul of he radom exerme are called elemeary eve. A e of all elemeary eve are called amle ace. Probably meaure real fuco defed uo a ube yem of he amle ace whch oegave, ormed ad σadve. Codoal robably a robably of a eve uder codo ha ome oher (o moble eve ha haeed. A ad B eve are deede f her ereco robably equal o a roduc of dvdual eve robable. Toal robably heorem gve u a way o deerme robably of ome eve A whle reumg ha comlee e of muual djo eve gve. Baye' heorem allow u o deerme codoal robable of dvdual eve h comlee e whle reumg ha A eve ha haeed. 53

Quz. How do you deerme he robably of wo eve uo?. How do you deerme he robably of wo eve ereco? 3. Whe are wo eve deede? Praccal Exerce Exerce : Suoe ha here a ma ad a woma, each havg a ac of 5 layg card. Each oe of hem draw a card from h/her ac. Fd he robably ha hey each draw he ace of club. {Awer: deede eve.37} Exerce : A gla jar coa 6 red, 5 gree, 8 blue ad 3 yellow marble. If a gle marble choe a radom from he jar, wha he robably of choog a red marble? a gree marble? a blue marble? a yellow marble? {Awer: P(red3/, P(gree5/, P(blue4/, P(yellow3/} Exerce 3: Suoe ha here are wo bowl full of cooe. Bowl # ha chocolae ch cooe ad 3 la cooe, whle bowl # ha of each. Fred c a bowl a radom, ad he c a cooe a radom. You may aume here o reao o beleve Fred rea oe bowl dfferely from aoher, lewe for he cooe. The cooe ur ou o be a la oe. How robable ha Fred ced ou of he bowl #? {Awer: Codoal robably.6} Exerce 4: Suoe a cera drug e 99% accurae, ha, he e wll correcly defy a drug uer a eg ove 99% of he me, ad wll correcly defy a ouer a eg egave 99% of he me. Th would eem o be a relavely accurae e, bu Baye' heorem wll reveal a oeal flaw. Le' aume a cororao decde o e emloyee for oum ue, ad.5% of he emloyee ue he drug. You wa o ow he robably ha, gve a ove drug e, a emloyee acually a drug uer. {Awer: Baye' heorem.33} 54

3. RANDOM VARIABLES Sudy me: 8 mue Learg objecve you wll be able o Decrbe he radom varable by he drbuo fuco Characerze a dcree ad a couou radom varable Uderad he hazard rae fuco Deerme he umercal characerc of he radom varable Traform he radom varable Exlaao 3.. Defo of a Radom Varable Le u coder a robably ace (Ω, S, P. A radom varable X (RV o a amle ace Ω a real fuco X(ω where for each real xî R he e { ω Î Ω X(ω < x } Î S,.e. a radom eve. Therefore, he radom varable a fuco X: Ω R where for each xî w Î W X(ω < x Î S. The defo mle ha we ca R hold: X ((, x { } deerme he robably of X( w < x for ay xîr. Samle ace R X W 55

A grou of all value { x X(ω w Î Ω} 3.. Drbuo Fuco, called amle ace. Defo: The drbuo fuco of a radom varable X F( ad for each ÎR ha he value: F( P{XÎ (, } P(X. Proere of he robably drbuo fuco:. F(x for < x < +. he drbuo fuco a moooc creag fuco of x,.e. " x, x Î R: x < x Þ F( x F( x 3. he drbuo fuco F(x lefcouou 4. lm F(x ; lm F(x x + x 5. " a, b Î R; a < b : P( a X < b F( b F( a 6. P( x x lm F(x F( x x x + If he rage of he radom varable fuco dcree, he he radom varable called a dcree radom varable. Oherwe, f he rage clude a comlee erval o he real le, he radom varable couou. 3.3. Dcree Radom Varable You ca ea abou dcree radom varable f a radom varable from ome fe ad eumerable e. The mo ofe a eger radom varable e.g. a umber of ude ha eered he ma buldg of VSB TUO before mdday (,,,..., a umber of houe occua (,,3,..., a umber of car accde o he Prague Bro hghway oe day (,,,..., ec.. Defo You ca ay ha a radom varable X ha a dcree robably drbuo whe: $ fe or eumerable e of real umber M{ x,..., x,... } ha P( X x >,,... P( X x Fuco P( X x Û P( x called robably fuco of radom varable X. A drbuo fuco of uch a drbuo a e fuco wh e x,..., x,.. 56

For a drbuo fuco of a dcree radom varable he followg rue: F( x P(X x < x x Examle Throwg a dce, X a umber of do obaed x P( X x F( x /6 /6 /6 3 /6 /6 4 /6 3/6 5 /6 4/6 6 /6 5/6 F(x P(x /6 /6 3 4 5 6 x 3 4 5 6 x 3.4. Couou Radom Varable If a radom varable ha ay value from a cera erval a radom varable wh couou drbuo. Produc lfe execacy (, or he legh a objec are examle. I h cae, a dey fuco a well a drbuo fuco ca be ued o decrbe a drbuo of radom varable. Defo Radom varable ha a couou robably drbuo whe a fuco f(x ex ha x ò F( x f ( d for < x < Fuco f(x called a robably dey fuco of couou radom varable X. I a oegave real fuco. I evde ha all o where a dervao of drbuo fuco ex he followg rue: 57

f ( x df( x dx If you ow he drbuo fuco you ca ealy deerme he robably dey fuco ad vce vera. The area below he f(x curve for x Î< a;b ; ( a,b Î R ay erval he robably ha X wll ge a value wh h erval. I alo fully correod wh he dey defo. P(a X < b F( b F( a f ( d f ( d f ( d b ò a ò b ò a P(a X < b f(x a b x Oe of he arbue of robably dey he fac ha he oal area uder he curve equal oe. I aalogcal o a dcree radom varable where he um of robable for all oble reul alo equal oe. The followg equao decrbe he arbue: ò f ( x dx 58

Examle Logc robably drbuo ha he followg drbuo fuco F(x ad robably dey f(x: F( x (β + βx f ( x (β + β x (β + βx +e β e ( + e.75.5 F(x.5 4 3 3 4.4.3 f(x.. 3.5.5.5.5.5.5.5 3.5 59

3.5. Falure Rae Le X be a oegave radom varable wh couou drbuo. The, he falure rae for F( < defed a: ( F( f l (. The followg formula ca ealy be derved: l( ( lm D + P < X + D X > f ( D F( Le X be a mea me o falure of ay yem. The, he falure rae mea ha f he me here wa o falure, he robably of falure a mall ubeque erval D aroxmaely l(.d : ( X + X P < D > f ( F( D l(.d The falure rae characerze he robably drbuo of oegave radom varable. Table how he muual covero bewee f (, F(, l ( : F( f( ( l F( F( ò f ( x dx é ù exê ò l( x dxú ëê ûú df( é f( l ex d ê ò l x dx ë df( f ( l( d l( F( ò f ( x dx f( ( ( ú û ù Table 6

The Mo Commoly Ued Grahcal Ierreao of Falure Rae Le a radom varable X be a mea me o falure of ay yem. The, a ycal form of falure rae how he followg fgure. The curve h fgure called he bahub curve. l ( I The fr ar a decreag falure rae, ow a early falure or fa moraly. II The ecod ar a coa falure rae, ow a radom falure. III The hrd ar a creag falure rae, ow a wearou falure. 3.6. Numercal Characerc of Radom Varable The robably drbuo of each radom varable X fully decrbed by drbuo fuco F(x. I may cae we ca ummarze he oal formao by everal umber. Thee umber are called he umercal characerc of he radom varable X.. Mome Rh Geeral Mome deoed m r ' EX r r,,, dcree RV: m r ' x r. P( x couou RV: m r ' ò xr. f( x dx r,,, f aed rogreo or egral coverge aboluely. Rh Ceral Mome deoed m r E[ X EX ] r r,,, 6

dcree RV: m r [ x EX ] r. P( x couou RV: m r ò ( x EX r. f( x dx f aed rogreo or egral coverge aboluely.. Execed Value (Mea EX m ' dcree RV: EX x. P( X x couou RV: EX ò x. f( x dx Proere:. E(aX + b a. EX + b a,b Î R. E( X + X EX + EX 3. X, X... deede RV Þ E( X. X EX. EX 4. Y g( X ; g( X a couou fuco: EY E( g( X Y a couou RV: EY ò g( x. f( x dx Y a dcree RV: EY g( x. P( X x 3. Varace DX m E( X EX EX ( EX dcree RV: DX couou RV: DX x.p( x ( x. P( x ò x. f( x dx ( ò x. f( x dx Proere::. D( ax + b a. DX. X, X... deede Þ D( X + X DX + DX 4. Sadard Devao x DX 5. Sewe a 3 m 3 / x 3 I a level of ymmery for he gve robably drbuo: a 3 ymmercal drbuo a 3 <... egavely ewed e 6

a 3 >... ovely ewed e 6. Kuro a 4 m 4 / x 4 I a level of uro (flae: a 4 3... ormal uro (.e. uro of ormal drbuo a 4 < 3... lower uro ha uro of ormal drbuo (flaer a 4 > 3... greaer uro ha uro of ormal drbuo (harer 7. Quale Î (, x... % quale x u{x½f(x } couou RV: F( x Secal ye of quale: x.5 5% quale called he meda x.5 ad x.75 5% quale called he lower quarle ad 75% quale called he uer quarle x /,, 9 he h decle x /,, 99 he h ercele 8. Mode The mode xˆ of a dcree RV X a value ha hold: P( X xˆ P( X x,,... I mea ha he mode a value whch he dcree RV come wh he hghe robably. The mode xˆ of a couou RV X a value ha mee he followg: f ( xˆ ³ f ( x ro < x < I a o where he robably dey ha he maxmum value. 63

4. RANDOM VECTOR Sudy me: 6 mue Learg Objecve you wll be able o Decrbe a radom vecor ad jo drbuo Exla he coce of margal ad codoal robably drbuo Exla a ochac deedece of radom varable Exlaao 4.. Radom Vecor For couou radom varable, he jo drbuo ca be rereeed eher he form of a drbuo fuco or a robably dey fuco. ( X < x, X < x,..., X F ( x <, x,..., x P x, F: R R f ( x, x,..., x F ( x, x,..., x x x... x Boh form are equvale. I erm of he jo robably dey fuco, he jo drbuo fuco of X,...,X x ò x ò (,,..., d d... d. F( x, x,..., x... f x ò Alhough heory he jo drbuo of a dcree varable wh a couou varable doe ex, here o raccal algebrac formulao of uch a drbuo. Thoe drbuo are oly rereeed codoal form. 64

4.. Margal Drbuo Defo Le X (X, X,..., X be a radom vecor. The radom vecor Y (X,X, ¼, X, where <, Î {,,..., }, ¹ ro u v, called he margal radom vecor. Eecally, X j u v ¹ he margal radom varable for every,,,. The robably drbuo of Y called he margal robably drbuo. Le X (X, X be a bvarae radom varable wh gve drbuo fuco F(x, x. F ( x lm F( x, x F( x x + x +, + margal drbuo fuco of radom varable X F ( x lm F( x, x F( +, x margal drbuo fuco of radom varable X For couou radom varable, he margal robably dey of oe joly drbued varable eablhed by egrag he jo dey fuco wh reec he oher varable. ò f ( x f ( x, x dx for X x ò f ( x f ( x, x dx for X. For dcree radom varable, he margal drbuo are gve by: P ( x P( X x, X x margal drbuo fuco of X x P ( x P( X x, X x margal drbuo fuco of X x x 4.3. Codoal Drbuo The codoal drbuo he drbuo of oe varable a a fxed value of he oher joly drbued radom varable. For wo dcree varable, he codoal drbuo gve by he rao of he jo robable o he correodg margal robably. ( x x P (X x X x P( X x, X P ( x x ( x, x P ( x. For couou radom varable, he codoal dee are gve aalogouly by he rao of he jo dey o he correodg margal dey. f(x x f ( x,x f ( x. 65

+ ( x ò f ( x,x dx f he correodg margal dey of X. 4.4. Ideedece of Radom Varable Defo X X are muually deede Û radom eve {X < x }, (,,,, where x Î R are muually deede. Therefore, X X are muually deede Û F(x,,x F (x.. F (x. The above rue becaue: F(x,,x P(X <x,, X <x P(X < x. P(X <x.. P(X <x F (x. F (x.. F (x. I mle he followg rule: X X are muually deede Û f(x,,x f (x.. f (x. Examle: X, X are muually deede. Deerme he varace X + X. I geeral, f X ad X are o deede, he varace of her um gve by D ( X + X D ( X + D ( X + Cov ( X, X where he covarace of X ad X defed by Cov ( X, X E [ ( X E ( X ( X E ( X ] Whe X ad X are deede, he covarace zero. A alerae exreo for he covarace mlar o ha of he varace ad mler for comuao Cov( X, X E( X X E( X E( X. Correlao Coeffce The correlao coeffce meaure he regh of he relao bewee wo radom varable, X ad X. The correlao coeffce defed by r XX Cov ( X, X σ σ X X. The correlao coeffce roere are:. ρ 66

. ρ(x,y ρ(y,x The correlao aume value bewee ad. A value cloe o mle a rog ove relaoh, a value cloe o mle a rog egave relaoh, ad a value cloe o zero mle lle or o relaoh. Examle ad Soluo Image ha you are gog o reea a e a form of a ral ad you are gog o reea hree me (you ow he ucce robably, mlarly a for hrowg a co Wre dow all oble combao: (S ucce, F falure: { FFF; SFS; SSF; FSS; FSF; FFS; SFF; SSS } Now you ca ecfy he followg radom varable: Y... a umber of aem ul he fr ucce Z... a umber of he ubeque uccee a deerme he robably fuco P( Y, P( Z b e he jo robably fuco Y, Z c deerme he margal drbuo fuco ad P( Y Z, P( Z Y Soluo: ada Y ad Z are he dcree RV ad ha why Y ad Z ca clam he value:,,, 3 All eleme eve of he amle ace eed o be gve her ame: A... FFF P( A ( 3 A... SFS P( A.( A3... SSF P( A3.( A4... FSS P( A4.( A5... FSF P( A5.( A6... FFS P( A6.( A7... SFF P( A7.( A8... SSS P( A8 3 The fac ha he F ad S varable are deede eed o be ae o coderao for he furher calculao: Y... a umber of aem ul he fr ucce 3 SFS, SSF, SFF, SSS FSS, FSF FFS FFF 67

Z... a umber of ubeque uccee 3 FFF SFS, FSF, FFS, SFF SSF,FSS SSS Sce A,..., A8 eve are djo we ca mly deerme he robably fuco (.5. Y... a umber of aem ul he fr ucce P(Y P(Y P( Y P(Y 3.5.5.5.5 Z... a umber of ubeque uccee P( Z P(Z P(Z P(Z 3.5.5.5.5 adb you roceed he ame way le eablhg he robably fuco Z Y 3 SFS, SFF SSF SSS FSF FSS FFS 3 FFF Z Y 3.5.5.5.5.5.5 3.5 adc margal robably fuco P( Y, P( Z Z Y 3 P( Y.5.5.5.5.5.5.5.5.5 3.5.5 P( Z.5.5.5.5 68

P( Y Z P( Y Ù Z / P( Z Z Y 3.5.5.5.5.5 3 P( Z Y P( Y Ù Z / P( Y Z Y 3.5.5.5.5.5 3 Summary Radom varable X a real fuco whch ca be characerzed by a drbuo fuco F(. Drbuo fuco a fuco ha ag o each real umber a robably ha he radom varable wll be le he h real umber. Drbuo fuco ha ome geeral roere le " a, b Î R; a < b : P( a X < b F( b F( a. Deedg o he ye of value he radom varable geg, here couou ad dcree varable. The dcree radom varable characerzed by a robably fuco, he couou varable by a dey fuco. Que ofe ueful o decrbe he whole formao abou radom varable by everal umber ha characerze roere whle allowg he comaro of dffere radom varable. Thee umber are called he umercal characerc of radom varable. A radom vecor a vecor cog of radom varable X (X, X,, X ha characerzed by jo drbuo fuco. From he jo drbuo fuco of he radom vecor you ca ealy deerme he margal robably drbuo of arcular radom varable he vecor comoed of. 69

Quz. Wha he muual relaoh bewee drbuo fuco ad robably fuco of a dcree radom varable?. Wha he muual relaoh bewee drbuo fuco ad robably dey fuco of a couou radom varable? 3. Wha are he meda ad he mode? 4. Exla he erm Codoal Probably Drbuo. 5. Exla he erm Sochac Ideedece of Radom Varable. 6. Wha doe a value of correlao coeffce ell u? Praccal Exerce Exerce : Le Y be a couou varable defed by a robably dey fuco: f( y c. ( + y.( y ; < y < elewhere Fd a coa c, a drbuo fuco, a execed value ad a varace of h varable. {Awer: c.75; F(y.5 (3y y 3 +; EY ; DY.} Exerce : Le a radom varable W be defed a a lear raformao of radom varable Y, defed revou examle. W 5Y + 6 Fd a robably dey fuco, a drbuo fuco, a execed value ad a varace of radom varable W. 3 {Awer: f(w 5 (w w+; F(w.5 [3( w 6 ( w 6 3 +]; EW 6; DW 5} 5 5 Exerce 3: Le a radom varable Z be defed a: f( z / [ ( + e z. ( + e z ]; < z < Fd a drbuo fuco of he radom varable Z. z e {Awer: F(z z } + e 7

5. SOME IMPORTANT PROBABILITY DISTRIBUTIONS 5.. Dcree Probably Drbuo Sudy me: 5 mue Learg Objecve you wll be able o Characerze Beroull ral ad ye of dcree drbuo Characerze Poo roce ad Poo drbuo Decrbe coexure bewee dcree drbuo Exlaao A lo of dcree radom varable ex ad ow we wll ummarze bac formao abou he mo commo dcree varable. Beroull Tral: a equece of Beroull ral defed a a equece of radom eve whch are muually deede ad whch have oly wo oble oucome (e.g. ucceoucce, robably of eve occurrece (a ucce coa ay ral P {Tral '' "Succe"} Bomal Radom Varable: The mo aural radom varable o defe o he amle ace of Beroull ral he umber of uccee. Such a radom varable called a bomal radom varable. If X he umber of uccee Beroull ral where he robably of ucce a each ral, he we reree he drbuo of X by he horhad oao: X B (, where B dcae ha X ha a bomal drbuo ad ad are he arameer deermg whch arcular drbuo from he bomal famly ale o X. 7

7 The robably drbuo of a bomal radom varable ca be exreed algebracally a: X P ø ö ç è æ ; ( ( X P EX (!!( (! ( (!! (! ( DX EX (EX EX EX X P EX + + + + ( ( (!!( (! ( ( (!! (! ( ( DX EX (EX..( Noce ha he varace of he bomal drbuo maxmum whe.5..5..5..5.5.5.75 ( Maxmum Examle: Some examle of bomal drbuo for 4 ral are lluraed below. Noce ha a, he robably of ucce a each ral creae, he locao of he drbuo hf o hgher value of he radom varable. Alo oce ha whe.5, he drbuo ymmerc aroud 7.5.

P ( X.3.5..5..5.5.85.5 3 4 5 6 7 8 9 3 4 Geomerc Drbuo: Th drbuo ha a gle arameer,, ad we deoe he famly of geomerc drbuo by X G ( G( he geomerc radom varable defed a he umber of ral ul a ucce occur or ul he fr ucce The robably drbuo for a geomerc radom varable : P( X ( ; < The exreo for he mea of he geomerc drbuo EX æ P( X ( ( ç ( ( è ö ø Noe: By fr evaluag he ere ad he ag dervave he reul obaed. The mea umber of Beroull ral ul he fr ucce he vere of he ucce robably a each ral, aga a erely uve reul. Tha, f % of he ral are ucceful, o average wll ae e ral o oba a ucce. To fd he varace, we fr evaluae he execed value of X ad he modfy he exreo ug he ame echque a for he bomal cae. We oe ha he exreo ow ha he form of he ecod dervave of he ame geomerc ere we evaluaed for he mea. Tag dervave of h evaluaed exreo, we oba 73

74 X E ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( + ø ö ç è æ + ø ö ç è æ + + From he mea ad execed value of X we ca derve he varace. ( EX EX DX Examle: Some examle of geomerc drbuo are lluraed below. No urrgly, he robably of log equece whou ucce decreae radly a he ucce robably,, creae. ( X P.5.5.75 4 6 8.85.5.5

75 Negave Bomal Radom Varable The egave bomal drbuo ha wo arameer, ad ad deoed by, ( NB X The egave bomal he umber of Beroull ral ul he h ucce. The egave bomal drbuo : < ø ö ç è æ X P ; ( ( The mea ad varace of he egave bomal drbuo ca be comued ealy by og ha a egave bomal radom varable wh arameer ad ju he um of deede geomerc radom varable wh arameer. Ideedece he geomerc radom varable follow from he deedece aumo of equece of Beroull ral. Thu f G W ; ( he ( ( ( ( ( W D X D W E X E W X

Examle: The followg char llurae ome examle of egave bomal drbuo for 3. Noce ha for value of ear.5, he drbuo ha a gle mode ear 5. Th mode move away from he org ad dmhe magude a decreae dcag a creae varace for mall. The egave bomal drbuo ha a hae mlar o he geomerc drbuo for large value of..7.6.5 P ( X.85 K 3.4.3..5.5. 3 8 3 8 3 8 Noe: Comaro of Bomal ad Negave Bomal Drbuo I ereg o comare he drbuo of bomal ad egave bomal radom varable. Noce ha exce for he combaoral erm a he begg of each drbuoal exreo, he oro corbued by he robably of gle amle ace eleme decally (. Bomal drbuo P( X æö ç ( è ø ; For he bomal, he umber of ral ( fxed ad he umber of uccee ( varable. Negave bomal drbuo P( X æ ö ç è ø ( ; < For he egave bomal, he umber of uccee ( fxed ad he umber of ral ( varable. 76

Poo Proce The Poo roce a ecod geeral ye of amle ace model whch wdely aled racce. The Poo roce may be vewed a he couou me geeralzao of a equece of Beroull ral, omeme called he Beroull roce. The Poo roce decrbe he amle ace of radomly occurrg eve ome me erval. The Poo roce aume ha he rae a whch eve occur coa hroughou he erval or rego of obervao ad hoe eve occur deedely of each oher. l lambda he rae a eve occur However, eve occurrg over ome rego ca alo be modeled by a Poo roce. The aearace of defec ome roduc, or mold o he urface of a leaf uder cera codo could follow a Poo roce. Examle: Cuomer arrvg a a ba o raac bue. Pae arrvg a a clc for reame Telehoe qure receved by a goverme offce, ec. Some examle of eleme of he amle ace for a Poo roce are lluraed below. Th a comlex dffcul o characerze amle ace. The umber of eleme he amle ace ucouable, fe ad h ee couou. Probable cao be aged o dvdual eleme of h amle ace, oly o ube. Samle # Samle # Samle #3 Tme of Occurrece The Poo roce decrbe eve whch occur radomly over ome me erval or aal rego. 77

78 Poo Drbuo The Poo drbuo ha a gle arameer ad herefore we deoe h radom varable by he ymbolc oao, ( P X l Coder a Poo roce ha oberved for a me erod. Suoe he rae of occurrece of eve l durg he me erod. The he oal rae of occurrece over he ere obervao erval l. Now dvde he erval o uberval of equal legh /. Occurrece of eve each of hee erval wll be muually deede a coa rae l/. If become large eough, he erval legh, /, wll become mall eough ha he robably of wo eve oe erval effecvely zero ad he robably of oe eve rooroal o l/. The he drbuo of he umber of eve he oal erval ca be aroxmaed by he drbuo of a bomal radom varable wh arameer ad l/. Thu, X P ø ö ç è æ ø ö ç è æ ø ö ç è æ l l ( Tag he lm a goe o fy, h exreo become,! e (! (! lm! ( lm l l l l l l ø ö ç è æ ø ö ç è æ ø ö ç è æ ø ö ç è æ We ca exre he drbuo of a Poo radom varable a: < e X P ;! ( ( l l! (! ( ( e e X P l l l l Calculao of mea:! ( e (! e ( X P( X E ( l l l l l l The execed value of X foud ug he Taylor ere exao of he exoeal fuco a well a he ame algebrac maulao a wa voed for Beroull radom varable.

79 e e X P X E l l l l l l l l + ( +! ( ( ( + E (X! ( ( ( ( From h reul, he varace follow drecly. X E ( X ( E X ( D l (E (X E (X ] [ We oce ha he Poo drbuo ha he remarable roery ha he varace equal o he mea ad by mlcao ha he varace of he Poo radom varable wll creae a he rae l creae. Examle: Some examle of Poo drbuo are lluraed below. Noce ha a he value l 9, he drbuo become almo ymmerc..5..5..5 3 4 5 6 7 8 9 3 4 5 6 7 8 9 l 3 l 5 l 9

5.. Couou Probably Drbuo Sudy me: 5 mue Learg Objecve you hould be able o Characerze ye of couou drbuo : exoeal, Gamma ad Webull Decrbe coexure bewee couou drbuo Exlaao How doe a bac decro for exoeal, gamma ad Webull drbuo loo le? Exoeal Drbuo The exoeal radom varable a ecod very aural varable whch ca be defed o he amle ace geeraed by a Poo roce. If a couou me roce afe he aumo of a Poo roce, he me bewee eve, or equvalely becaue of he aumo of deedece, he me ul he ex eve wll have a exoeal drbuo. The exoeal radom varable aalogou o he geomerc radom varable defed for a Beroull roce. The rage of oble value for he exoeal radom varable he e of oegave umber. T Tme Bewee Eve Tme of Occurrece T ha a Exoeal Drbuo 8

Srcly eag, he amle ace for a exoeal radom varable co of erval of varyg legh ermaed by a gle eve, ju a he amle ace of he geomerc radom varable co of a equece of falure ermaed by a ucce. The robably dey fuco ad drbuo fuco of a exoeal drbuo have he followg mle form. l f ( le ; ³ F(P(T < P(N ³ P(N < where l he rae a whch eve occur. The famly of exoeal radom varable defed by he gle arameer, l, he ame arameer whch defe he Poo drbuo. e l T E (l The mea of he exoeal drbuo he recrocal of he rae arameer. The reul ca be obaed hrough egrag he execed value egral by ar. E ( T ò l e d l l The varace of he exoeal drbuo obaed from evaluag he followg egral aga hrough egrao by ar. DT ET ET (... l The varace equal he quare of he mea ad herefore he mea equal he adard devao for a exoeal drbuo. The hazard fuco gve by: f ( h( f F( < F( h( l co. Þ he "o memory" roery of he exoeal drbuo Examle: The followg grah llurae ome examle of he robably dey fuco of exoeal radom varable. Noce ha he hae of he exoeal dey mlar o he hae of he geomerc robably drbuo. The exoeal drbuo of me o ex eve he couou me equvale of he geomerc drbuo whch he umber of ral o ex eve where eve may be codered a "ucceful" ral. 8

9 8 7 6 5 4 3 f ( l 9 l 5 l 3.5.5.75 Gamma Drbuo T Ga (, l The amle ace geeraed by he Poo roce gve re o a ecod radom varable cloely aocaed wh he exoeal radom varable. The oal me ul ome ecfed umber of eve, ay, occur called a Gamma radom varable ad are a a um of decal deede exoeal radom varable. If he exoeal drbuo he couou me equvale of he geomerc, he follow ha he Gamma drbuo he couou me equvale of he egave bomal. T Tme Ul 4h Eve T ha a Gamma Drbuo Tme of Occurrece 3 4 5 The amle of a gamma radom varable arg a he um of 4 deede exoeal radom varable, ha a he me ul he fourh eve a Poo roce wll co of erval of varyg legh, all havg hree eve ad ermaed by a fourh eve. The gamma drbuo fuco for ay eger value of ca be derved by he followg argume. Sce he gamma are a he um of deede, decally drbued exoeal radom varable, he drbuo fuco of he gamma he robably ha 8

83 he um of exoeal le ha or equal o ome value. Th mle ha here have bee a lea occurrece of a Poo roce wh me, he robably of whch gve by he cumulave drbuo of a Poo radom varable wh rae arameer l, where l he rae of he uderlyg Poo roce. X X X X T... 3 + + + + ( l E X ( ( ( úû ù ê ë é < ³ ø ö ç è æ < <!! ( ( ( j j j j j e j e N P N P X P P T F l l l l ad he robably dey fuco ( ( ( ( ;!!! ( > ú û ù ê ë é ú û ù ê ë é ú û ù ê ë é e j e j e f j j j j l l l l l l l l Sce he gamma radom varable he um of decal deede exoeal radom varable, he mea ad varace wll be me he mea ad varace of a exoeal radom varable. Th ame argume wa ued o derve he mea ad varace of he egave bomal from he mome of he geomerc drbuo. l l l EX EX EX ET + + + + +......... l DT The form of he gamma drbuo reeed here where he arameer rerced o be a ove eger acually a ecal cae of he more geeral famly of gamma drbuo where a hae arameer whch eed oly be a ove real umber. The ecal cae we have dcued omeme called he Erlag Drbuo. Examle: Examle of gamma robably dey fuco for l are lluraed he followg char. Noce ha he gamma dey ha a gle mode whch move away from he org a creae. Alo he dero creae ad he drbuo become more ymmerc ad creae.

..9.8.7.6.5.4.3.. 3 5 7 4 6 8 4 6 8 The hazard fuco gve by: h l (! ( j! ( x j ( lx.9.8.7 3.6 5 7.5.4.3.. 4 6 8 4 6 The hazard fuco of Gamma drbuo, λ j h(x a harly creag fuco for > Þ h drbuo uable for modelg of ageg ad wear rocee 84

Webull Drbuo The drbuo fuco : F( x e æ x ö ç è Q ø b, Q >, b >, x > b... a hae arameer, Q... a cale arameer. The robably dey fuco for he Webull : Ad he hazard fuco for he Webull : f ( b æ x ö x ç Q èq ø b b e b æ x ö ç è Q ø b æ x ö h ( x. ç Q èq ø Some examle of he Webull dey ad he Webull hazard fuco are lluraed below. f(x b.5 Q b 3. b.5 b. 5 4 3 3 4 h(x Q b 3. b. b.5 b. b.5 3 4 85

The Webull drbuo very flexble ad we ue Relably heory for modelg of he radom varable "me o falure". Summary A equece of Beroull ral defed a a equece of radom eve whch are muually deede ad whch have oly wo oble oucome (e.g. ucceoucce, ad he robably of eve occur (a ucce coa ay ral. O he ba of hee ral execao we ca defe he followg radom varable: bomal, geomerc ad egave bomal. A umber of eve occurrece o ay deermc erval from o ca be decrbe (a cera execao by a Poo drbuo. If a couou me roce afe he aumo of a Poo roce, he me bewee eve, or equvalely becaue of he aumo of deedece, he me ul he ex eve wll have a Exoeal drbuo. A Gamma drbuo decrbe a me o h eve occurrece. A Webull drbuo geeralzao of he exoeal drbuo ad very flexble. Quz. Whch dcree ad couou drbuo do you ow?. Characerze he Beroull ral ad dvdual ye of he dcree drbuo. Deerme he mea of he bomal radom varable. 3. Wha Gamma drbuo ued for? How relaed o exoeal drbuo? 4. For wha b of Webull drbuo he hazard fuco learly creag? 86

Dagram: A Coexure Amog Drbuo Dcree Proce Couou Po Proce Beroull Tral Poo Proce Bomal Poo Number of Occurrece a Fxed Number of Tral Number of Occurrece a Fxed Perod of Tme Geomerc Exoeal Number of Tral o Fr Occurrece Tme o Fr Occurrece Negave Bomal Gamma Number of Tral o Kh Occurrece Tme o Kh Occurrece 87

Praccal Exerce Exerce : Suoe ha a co wh robably of head.4 hrow 5 me. Le X deoe he umber of head. a Comue he dey fuco of X exlcly. b Idefy he mode. c Fd P(X > 3. æ 5 ö {Awer: Le f( P(X ç (.4 (.6 5 for,,, 3, 4, 5. è ø a f(.778, f(.59, f(.3456, f(3.34, f(4.768, f(5.. b mode: c P(X > 3.987. } Exerce : Suoe ha he umber of mr N o a web age ha he Poo drbuo wh arameer.5. a Fd he mode. b Fd P(N > 4. {Awer: a mode:, bp(n > 4.88} Exerce 3: Meage arrve a a comuer a a average rae of he 5 meage/ecod. The umber of meage ha arrve ecod ow o be a Poo radom varable. a Fd he robably ha o meage arrve ecod. b Fd he robably ha more ha meage arrve a ecod erod. 7 {Awer: a 3.6(, b. 885} Exerce 4: If here are 5 cuomer er eghhour day a checou lae, wha he robably ha here wll be exacly 3 le durg ay fvemue erod? {Awer: Poo.88 } 88

6. THE NORMAL DISTRIBUTION AND THE LIMIT THEOREMS Sudy me: 6 mue Learg Objecve you wll be able o Characerze he ormal ad he adard ormal drbuo Formulae ad ue he lm heorem Decrbe ecal drbuo Exlaao 6.. Normal Drbuo Aroomer were reoble for oe of he earle aem o formally model he radom varao here he meaureme roce. The robably dey fuco whch wa adoed a ha me ha bee aleravely called he error fuco, he Gaua curve, ad oday mo commoly, he ormal drbuo. The ormal drbuo he mo wdely ued model of radom varao. I oulary arly baed o uve aeal a a mle mahemacal model of our cual oo of wha coue radom varao. However, here alo oud heorecal uor for he belef ha he ormal drbuo frequely occur racce. The form of he ormal dey model a mle ymmerc bellhaed curve wh a gle mode. Th hae acheved by he ue of a egave exoeal fuco whoe argume he quare of he dace from he mode. Sce quared dace mae value ear zero maller, he ormal curve ha a mooh rouded hae he rego of mode. The ormal dey ha wo arameer, µ, mode ad he o abou whch he dey ymmerc, ad σ, a cale or dero arameer whch deerme he cocerao of he dey abou he mode ad he rae of decreae of he dey oward he al of he drbuo. The famly of ormally drbued radom varable deoed by X N( m, The robably dey of he radom varable X wh he ormal drbuo: X N( m, 89

f ( x e æ x m ç è ö ø ; < x < + ro < µ <, > Th dey ymmerc abou µ ad herefore he mea, meda, ad mode are all equal o µ. Alo due o he ymmerc bell hae of h dey, he erquarle rage equal he Shorh whch wce he MAD. The followg char llurae he drbuo of robable for a ormal radom varable. The fr char how ha he robably of beg bewee ad adard devao (σ above he mea (µ.343 or aroxmaely oehrd. Sce he drbuo ymmerc, he robably of beg bewee ad adard devao (σ below he mea (µ alo aroxmaely oe hrd. Therefore he robably of beg more ha oe adard devao from he mea eher dreco aga oe hrd..4 f (x P.343.3. P.359. P.75 m3 m m m m+ m+ m+3 x Coverely, he uer ad lower quarle are wo hrd of a adard devao above ad below he mea. Thu µ +/.67 σ dvde he robably of he drbuo o four equal ar of 5%. 9

.4 Lower Quarle µ.67σ Uer Quarle µ+.67σ.3.. µ3 µ µ m µ+ µ+ µ+3 The mea ad varace of a ormal radom varable are equal o locao arameer µ, ad he quare of cale arameer σ, reecvely. E( X + ò x e æ xm ö ç è ø V ( X E[( X E( X ] dx m + ò ( x m e æ xm ö ç è ø dx The drbuo fuco of he ormal drbuo : x F( x ò f ( d ò e x ( m d d Sadard Normal Drbuo A ormal radom varable wh locao arameer ad cale arameer called a adard ormal radom varable. Becaue of he form of he ormal dey, oble o deerme robable for ay ormal radom varable from he drbuo fuco of he adard ormal varable. Coequely, he adard ormal radom varable ha bee gve he ecal ymbolc deao, Z, from whch he zcore derve. The adard ormal drbuo fuco gve he ecal ymbol, Φ. The, Φ( z P( Z < z z ò j ( u du z ò e u du 9

where φ(z he adard ormal dey j ( z e z ; < z < + Sadard Normal Drbuo 4 3 3 4 Sadardzao relao bewee ormal ad adard ormal drbuo Therefore f X ay ormal radom varable N(m, we ca defe a relaed adard ormal radom varable Z X m ad ha he adard ormal drbuo. X... N(m, X m Þ Z, Z... F(, The drbuo fuco of X ca herefore be comued from he derved radom varable Z whch ha a adard ormal drbuo: x x ( m d xm u F( x P( X < x ò f ( d ò e d ò e du F( xm EXAMPLE AND SOLUTION X... N(, 5, deerme P( < X < 8 Soluo: P( < X < 8 F( 8 F( F( 8 F( F(. F(.885.5.385 5 5 9

I he able or by uable ofware we ca fd: F(..885, F(.5 Examle: The followg char llurae he ormal dey wh zero mea for eleced value of. I clear ha he mea, meda, ad mode of a ormal radom varable are all equal, ad he wo arameer of he ormal drbuo are he embodme of our uve oo abou he geeral drbuoal characerc of locao ad cale. Sadard ormal j ( z.5 6 4 4 6 6.. Lm Theorem Bac defo Covergece robably: lm P( X X < e Þ X coverge robably o radom varable X ¾ X;.e. a equece of radom varable {X } Covergece drbuo: {F (x}... a equece of drbuo fuco correodg o radom varable {X } The equece {X } coverge oward X drbuo, f: lm F ( x for every real umber x a whch F couou. Coequece: F(x, 93

The equece {X } coverge drbuo o drbuo N(m,,.e. he radom varable x µ X ha aymocal ormal drbuo f lm F ( x Φ(. σ Lm Theorem Chebyhev' Iequaly: X... a arbrary radom varable wh mea EX ad varace DX. The P( ôx EX ô ³ e DX e, e > Th relao reul from he varace defo: D(X ³ + ò ( xe(x f ( x dx ( xe(x f ( x dx + ( xe(x f ( x ò xe( X ³ e ïì í ïî xe ( X < e ( xe(x f ( x dx ³ e P( X E( X ³ e ò ò xe ( X ³ e ïü dxý ïþ Ug he Chebyhev' equaly (for calculag robable: e.g. we ca aly o P( X E ( X < > X X wh reec o followg lm heorem: P( X m < > X m P( < > Law of Large Number: {X }... a equece of deede radom varable, each havg a mea EX µ ad a varace DX σ. Defe a ew varable X. X j j, Î N The equece { X } coverge robably o µ: X ¾ µ. Noo: Th affrmao reul from he Chebyhev' equaly. 94

Beroull heorem: {X }... a equece of he bomal deede radom varable wh arameer a Î (, (ocalled alerave radom varable, le X cae he eve wll be a oe ral ad X cae he eve wo' be; P(X a P(X. The we ow ha. ¾ ¾ X X j j The exreo o he lef de reree a relave frequecy of he eve occurrece he equece ral. Tha why we ca emae a robably gog ay occurrece by relave frequecy of h eve occurrece he equece ral whe we have a grea umber of he ral. Ceral Lm Theorem Ldeberg' Theorem: Le X, X,..., X... be a equece of deede radom varable,. X... ha he ame robably drbuo, EX m, DX. The X m Y ha a aymoc ormal drbuo N(, Þ lmp( Y u < F(u for < u <, mea ha Y coverge drbuo o drbuo N(,. For uffcely large umber he followg rue:. X X Þ EX m, DX, we ca aroxmae he drbuo drbuo N( m,,.e. N( m,.. Aalogouly for X : X X X ha he aymoc ormal drbuo, X by he X X ha he aymoc ormal drbuo wh arameer E X m, D X, X N( m, A ecal cae of he above heorem he MovreLalace Theorem: Le S... B(, ; ES ; DS ( 95

96 [ S X, X... ha he alerave drbuo hu bomal B(, ] he for large hold ha (, ( N S U. Alcao of he Ceral Lm Theorem Normal Aroxmao o he Bomal ad Poo drbuo Tag obervao of a Beroull drbuo ad comug he amle average ˆ equvale o defg he amle rooro radom varable X ˆ... rooro of uccee Beroull ral The amle rooro wll have a Bomal drbuo wll he value recaled from o. Tha, f X ha a bomal drbuo wh arameer ad. X ø ö ç è æ ú û ù ê ë é ( ] P [ ˆ P Furhermore: D E ( ˆ ( ; ˆ ( Therefore by he Ceral Lm Theorem, we ca aroxmae he bomal drbuo by he ormal drbuo for large. (, ( ˆ N (, ( N X Probable of value rage for hee varable ca he be calculaed a ø ö ç ç ç ç è æ ø ö ç ç ç ç è æ < < Φ Φ P ( ( ˆ ( S X

97 ø ö ç ç è æ ø ö ç ç è æ < < ( ( ( Φ Φ S P For maller amle ze a ocalled couy correco ofe emloyed o mrove he accuracy of he aroxmao. Thu we would comue he recedg robably a ø ö ç ç è æ ø ö ç ç è æ + < < (,5 (,5 ( Φ Φ S P The Ceral Lm Theorem ale broadly o mo drbuo. I arcular, he ormal drbuo ca be ued o aroxmae he Poo drbuo whe he erval of obervao,, ad hece he execed umber of eve, l, large. ˆ occurrece of rae me u er cou X l We ow ha he mea ad varace of he umber of eve durg a erval l, ad herefore he mea ad varace of he rae a whch eve occur X D X E l l ( ; ( Probable cocerg Poo cou or rae ca he be calculaed a ø ö ç è æ ø ö ç è æ < < Φ Φ X P l l l l ( ø ö ç ç ç ç è æ ø ö ç ç ç ç è æ < < g Φ g Φ g X g P l l l l ( where, g g ; Alyg he couy correco, we would calculae he robably a, ø ö ç è æ ø ö ç è æ < < Φ Φ X P l l l l,5 +,5 (

98 6.3. Secal Samlg Drbuo ChSquare Drbuo The chquared radom varable are a he um of quared adard ormal radom varable. The drbuo ha a gle arameer, he umber of quared ormal radom varable he um. Th arameer called he degree of freedom of he chquared drbuo. Z c A cquared radom varable wh oe degree of freedom c mly a quared adard ormal radom varable. The drbuo fuco of h radom varable ( ( ( ( ( > < < < y y Φ y Φ y Z y P y Z P y F c The robably dey fuco of a c radom varable ca be foud from he dervave of drbuo fuco. > ø ö ç ç è æ + y y e y y y y ( y y ( y y ( F y ( f y F F c c The geeral dey for a c radom varable ø ö ç è æ G ø ö ç è æ ( e x x f x c where ( G a gamma fuco. The mea ad varace of he chquared drbuo are D E ( ( c c The dey fuco for varou value of he arameer :

.5.4.3. 4 8. 4 6 8 The c he amlg drbuo of he amle varace ( x x If he X a amle from a ormal oulao wh mea m ad adard devao, he he amle varace ha a drbuo c To ee h, coder he um of quared adardzed obervao from a ormal oulao wh mea m ad adard devao. æ x m ö ç è ø c Th exreo clearly ha a c drbuo. Now reexre he umeraor by he added ad ubracg he amle mea from he quared erm. Exadg ad mlfyg, we oba ( x x + x m c 99

( x x ( x m + c The ecod erm mly he quared adardzed amle mea ad herefore ha a c drbuo. Sce he c he um of quared ormal, he fr erm mu be he remag quared ormal. Therefore, ( x x ( c Noe ha h argume oly heurc ad o a formal roof. The deedece of he amle mea from he devao abou he amle mea ha o bee eablhed. Th fac mora a he acal hyohe eg.. We ue h drbuo for verfcao of he radom varable deedece.. We ca ue chquare drbuo o e ha he radom varable follow cera drbuo. Th e ow a "GoodeofF Te". Sude Drbuo (drbuo The Sude' drbuo he amlg drbuo of he adardzed amle mea whe he amle varace ued o emae he rue oulao varace. The org of ame ha a ereg hory. A Irh aca, W. S. Goe fr ublhed h drbuo aoymouly uder he eudoym "Sude" becaue h emloyer, Gue Brewere of Dubl, rohbed emloyee from ublhg uder her ow ame for fear ha comeor would dcover he ecre of her excelle beer. I h orgal aer, Goe ued he degao "" for h ac. Hece he ame Sude'. The Sude' drbuo wh degree of freedom he rao of a adard ormal radom varable o he quare roo of a chquared radom varable dvded by degree of freedom. The drbuo ha a gle arameer, he degree of freedom of he chquared radom varable he deomaor. Z c The robably dey fuco of h radom varable f ( æ + ö Gç è ø æ ö ç + æ ö Gç è ø è ø +

The mea ad he varace of he drbuo are E ( D( The followg fgure how he dey fuco for dffere value of he degree of freedom:.4.3 6 4.. 6 4 4 6 If radom varable X, X,..., X have he ormal drbuo N(µ, ad hey are deede he X N( m, ad X m N(, he X m S æ ö ç X m ç ç è ø S ( Sude' drbuo ha a wde uage

FherSedecor Drbuo Fdrbuo Sedecor' Fdrbuo are a he rao of wo chquared drbuo dvded by her reecve degree of freedom. The Fdrbuo ha wo arameer, he degree of freedom of he chquared radom varable he umeraor ad m he degree of freedom of he chquared radom varable he deomaor. c F, m c m The robably dey fuco of h radom varable m æ + m ö Gç è ø m f ( x + m æ ö æ m ö m x Gç Gç ( m + x è ø è ø The mea ad varace of he drbuo are æ m ö m ç+ m (, (, è E F ø m D F m m ( m ( m 4 The followg fgure how he dey fuco for dffere value of m a :.3.5. 3 m 3 6 m.5..5 3 m 5 4 m 8 4 6 8 Clearly h drbuo would are a he amlg drbuo of he rao of he amle varace of wo deede oulao wh he ame adard devao. / The degree

of freedom reree oe le ha he amle ze of he umeraor ad deomaor amle varace reecvely. X; X X X j j N( m, ; N( m, ; j, j, S j ( X X j ;, S S F, We ue h drbuo for evaluao of acal aaly reul. Summary Oe of he mo mora couou drbuo he ormal drbuo. I a drbuo wh wo arameer he mea ad he varace. The adard ormal drbuo achevable wh eleced arameer (whe he mea equal ad he varace equal. Chebyhev' equaly u a uer boudary o he robably ha a obervao hould be far from mea. Chíquare drbuo a drbuo derved from he um of quared adard ormal radom varable. Ceral lm heorem decrbe aymoc acal behavor of he mea. We ca ue for ubug he bomal (Poo drbuo by he ormal drbuo. Sude' drbuo wh degree of freedom he rao of he adard ormal radom varable o he quare roo of he chquared radom varable dvded by degree of freedom. Fdrbuo he rao of wo chquared drbuo dvded by her reecve degree of freedom. Quz. Defe he relaoh bewee he ormal ad he adard ormal drbuo.. Wha he Chebyhev' equaly? 3. Exla he law of large umber. 3

4. Decrbe he chquare drbuo. Praccal Exerce Exerce : If he mea (m hegh of a grou of ude equal 7cm wh a adard devao ( of cm, calculae he robably ha a ude bewee 6cm ad 8cm. {Awer:,688} Exerce : Le X be he "hegh of a radomly eleced male", ad uoe ha X ormally drbued wh µ 76cm ad 5 cm,.e. X ha N(76,5: ( calculae he robably ha he hegh of a radomly eleced male le ha or equal o 8cm ( calculae he robably ha he hegh of a radomly eleced male le ha or equal o 7cm ( calculae he robably ha he hegh of a radomly eleced male o greaer ha 76cm (v calculae he robably ha he hegh of a radomly eleced male bewee 7 ad 8 cm (v calculae he robably ha he hegh of a radomly eleced male o le ha 6cm {Awer: (.885, (.5, (.5, (v.7698. (v.9993} Exerce 3: You ae he hegh of 9 male ad you aume ha he hegh are N(76,5 a he revou exerce. Wha he robably ha he amle mea hegh bewee 74cm ad 78cm? {Awer:.7698} Exerce 4: Premaure babe are hoe bor more ha 3 wee before he due dae. A local ewaer reor ha % of he lve brh a coury are remaure. Suoe ha 5 lve brh are radomly eleced ad he umber Y of "reeme" deermed. ( Wha he robably ha X le bewee 5 ad 3 (boh cluded? ( Fd he rooro of he eve ha fewer ha brh are remaure? {Awer: (.86, (.} 4

7. INTRODUCTION TO STATISTICAL INFERENCE Sudy me: 5 mue Learg Objecve you wll be able o Uderad he radom amlg erm Ue he amlg drbuo ad her roere Exlaao 7.. Iroduco The Scefc Mehod Scece a roce of yemac learg whch roceed by alerag bewee ducve ad deducve mehod of vegao. Deduco THEORY Iduco DATA The mehod of duco ad deduco are he coeco bewee daa ad heory of a cece. The deducve mehod roceed a logcally coe faho o rojec wha daa hould reul from a arcular heory. Iduco a formal roce whch re o oulae ome heory o reaoably exla he oberved daa. Sac o be a comlee cece mu embody boh ducve ad deducve mehod. The fr oc of he coure, exloraory daa aaly, wa a aem o uderad oberved drbuo of daa ad wa herefore a ducve mehod. Whou ome heory of radome however, he ably of EDA mehod o duce rece 5

exlaao wa lmed. Therefore, we roduced he heory of robably ad dcued alcao o varou hyohecal amle ace. Probably heory he ba of deducve mehod of ac. Probably heory roceed by aumg a hyohecal amle ace or oulao o whch a robably meaure defed. Probably drbuo of radom varable defed o h amle ace are he derved mahemacally. The robably of ay amle obervao from he hyohecal oulao ca he be deermed. Deducve Theory of Probably POPULATION ( W Samle Kow Probably Drbuo of Poulao Samlg Drbuo of Samle Sac If robably heory he deducve mehod of ac, he by mlcao, heory acal cece mu be rereeed by ome welldefed oulao wh a ow robably drbuo ad daa by he amle draw from ha oulao. Sacal ferece he become he ducve mehod for ug amle daa o mae ferece abou he robably drbuo of he oulao from whch he amle wa draw. Poulao Probably Theory (Deduco Sacal Iferece (Iduco Samle Sacal ferece he he vere of he robably heory. I he roce of mag aeme abou a uow oulao o he ba of a ow amle from ha oulao. 6

7.. Radom Samlg I mo ac roblem, we wor wh a amle of obervao eleced from he oulao you are ereed. The followg fgure llurae he relaoh bewee he oulao ad he amle. We have formally dcued hee coce before, however, we are ow gog o ree formal defo for ome of hoe erm. Defo: A oulao co of he oaly of he obervao wh whch we are cocered. I ay arcular roblem, he oulao may be mall, large bu fe, or fe. The umber of obervao he oulao called he ze of he oulao. For examle, he umber of udefled bole roduced o oe day by a ofdr comay a oulao of fe ze. The obervao obaed by meaurg he carbo mooxde level every day a oulao of fe ze. We ofe ue a robably drbuo a a model for a oulao. For examle, a rucural egeer mgh coder he oulao of ele regh of a cha rucural eleme o be ormally drbued wh mea ad varace. We could refer o h a a ormal oulao or a ormally drbued oulao. I mo uao, moble or mraccal o oberve he ere oulao. For examle, we could o e he ele regh of all he cha rucural eleme becaue would be oo mecoumg ad exeve. Furhermore, ome (erha may of hee rucural eleme do o ye ex a he me a deco o be made, o o a large exe, we mu vew he oulao a coceual. Therefore, we deed o a ube of obervao from he oulao o hel mae deco abou he oulao. Defo: A amle a ube of obervao eleced from a oulao. 7

For acal mehod o be vald, he amle mu be rereeave of he oulao. I ofe emg o elec he obervao ha are mo covee a he amle or o exerce judgme amle eleco. Thee rocedure ca frequely roduce ba o he amle, ad a a reul he arameer of ere wll be coely uderemaed (or overemaed by uch a amle. Furhermore, he behavor of a judgme amle cao be acally decrbed. To avod hee dffcule, derable o elec a radom amle a he reul of ome chace mecham. Coequely, he eleco of a amle a radom exerme ad each obervao he amle he oberved value of a radom varable. The obervao he oulao deerme he robably drbuo of he radom varable. To defe a radom amle, le X be a radom varable ha reree he reul of oe eleco of a obervao from he oulao. Le f(x deoe he robably dey fuco of X. Suoe ha each obervao he amle obaed deedely, uder uchagg codo. Tha, he obervao for he amle are obaed by obervg X deedely uder uchagg codo, ay, me. Le X deoe he radom varable ha reree he h relcae. The, X, X X a radom amle ad he umercal value obaed are deoed a x, x,,x. The radom varable a radom amle are deede wh he ame robably drbuo f(x becaue of he decal codo uder whch each obervao obaed. Tha, he margal robably dey fuco of X, X X f (x, f (x, f (x reecvely, ad by deedece he jo robably dey fuco of he radom amle f XX, X (x, x,,x f(x f(x f(x. Defo: The radom varable X, X X are a radom amle of ze f (a he X are deede radom varable, ad (b every X ha he ame robably drbuo. To llurae h defo, uoe ha we are vegag he effecve ervce lfe of a elecroc comoe ued a cardac acemaer ad ha comoe lfe ormally drbued. The we would exec each of he obervao o comoe lfe a radom amle of comoe o be deede radom varable wh exacly he ame ormal drbuo. Afer he daa are colleced, he umercal value of he oberved lfeme are deoed a x, x,,x. The rmary uroe ag a radom amle o oba formao abou he uow oulao arameer. Suoe, for examle, ha we wh o reach a cocluo abou he rooro of eole he Ued Sae who refer a arcular brad of of dr. Le reree he uow value of h rooro. I mraccal o queo every dvdual he oulao o deerme he rue value of. I order o mae a ferece regardg he rue rooro, a more reaoable rocedure would be o elec a radom amle (of a arorae ze ad ue he oberved rooro ˆ of eole h amle favorg he brad of of dr. The amle rooro, ˆ comued by dvdg he umber of dvdual he amle who refer he brad of of dr by he oal amle ze. Thu, ˆ a fuco of he oberved value he radom amle. Sce may radom amle are oble from a 8

oulao, he value of ˆ wll vary from amle o amle. Tha, ˆ a radom varable. I called a ac. Defo: A ac ay fuco of he obervao a radom amle. We have ecouered ac before. For examle, f X, X X a radom amle of ze, he amle mea X he amle varace S, ad he amle adard devao S are ac. Alhough umercal ummary ac are very ueful, grahcal dlay of amle daa are a very owerful ad exremely ueful way o vually exame he daa. I he fr lecure we reeed a few of he echque ha are mo releva o egeerg alcao of robably ad ac. 7.3. Samlg Drbuo Le aume ha gve radom amle come from ormal drbuo: X(X,..., X, X N( m, X æ. ö X N ç m, come from ceral lm heorem for large umber è ø X m. Z N(, come from a raformao of revou drbuo S 3. ( c ( wa exlaed c dcuo where S ( X X S ; ( X æ ö X X ( ç X è ø X m 4. drbuo ce: wa derved he dcuo abou ug of Sude X m S ( X m S Now aume wo amle from he ormal drbuo X(X,..., X, X N( m,, Y(Y,..., Y m, Y j N( m,. The hold: 9

5. (, ( N m Y X + m m 6., ( ( m y x y x F S S m m S S exlaed F drbuo Now aume ha he varace are he ame ad are uow:. The he followg roved o be rue: 7. (.. ( ( ( + + + + m y x m m m m S S Y X m m Summary The radom amle he ecal radom vecor whoe eleme are deede radom varable wh he ame robably drbuo. If he radom amle come from he ormal drbuo of robably we ca derve oher gfca ac wh ow drbuo from gve radom amle, e.g. ac X m or woamle ac: (.. ( ( ( + + + + m y x m m m m S S Y X m m. Thee oher ac wll be laer ued for he coruco of erval emao or for hyohe eg.. Wha he acal duco?. Characerze he erm: radom amle. ø ö ç ç è æ + ø ö ç ç è æ ø ö ç ç è æ m N Y X m N Y N X, ~, ~, ~ m m m m Quz

8. HYPOTHESIS TESTING Sudy me: 8 mue Learg Objecve you wll be able o Coclude by he ure gfcace e Ue bac amle ad wo amle e Coclude by ared e ad e for rooro Exlaao 8.. Iroduco Th chaer gog o deal wh corucg a e ha ued a a ad acceg or rejecg ome oulao hyohee. The mo commo uao whe he oulao ca be decrbed by ome robably drbuo whch deed o θ arameer. Baed o he ral reul we ca acce or rejec a oo ha θ ha ome cocree value θ. I oher uao ad for hyohe valdy we wll be ereed wheher he gve oulao come from a cera drbuo. Procedure leadg o mlar deco are called gfcace e. Sacal hyohe he aumo abou oulao of whch ruee ca be verfed by acal gfcace e Sgfcace e rocedure whch decde f a verfed hyohe hould be acceed or rejeced baed o a radom amle Null hyohe H a verfed hyohe of whch rejeco decded by he gfcace e Alerave hyohe H A a hyohe whch acceed whe he ull hyohe rejeced 8.. Pure Sgfcace Te The ure gfcace e a wheher he amle reul exreme wh reec o ome hyohezed drbuo.

If he amle daa le a a exremely hgh or exremely low ercele of he hyohezed drbuo, he he hyohe doub. HYPOTHESIZED POPULATION Kow amle Daa I daa coe wh hyohezed oulao? The ure gfcace e co of he followg comoe:. Null Hyohe: H The ull hyohe exree ome belef abou he aure of he oulao. I mu be ecfed recely eough o defe a robably meaure o he oulao.. Samle Sac: T(X The amle ac a fuco of he amle daa draw from he oulao. The choce of amle ac deermed by he characerc of he oulao' robably drbuo wh whch he ull hyohe cocered. 3. Null Drbuo: F(x F ( x P( T( X < x H The ull drbuo he robably drbuo of he amle ac whe he ull hyohe correc. The ull hyohe mu be ecfed recely eough o deerme he ull drbuo. 4. To deerme wheher he oberved amle ac x OBS exreme wh reec o he ull drbuo, a ac ow a he value comued. The value ha 3 defo deedg o he coex of he ull hyohe, bu all cae, he erreao of he value he ame. Defo : P VALUE F (x OBS Th defo ued whe we are cocered ha he drbuo of he amle ac may be le ha he ull drbuo.

Defo : P VALUE F (x OBS Th defo ued whe we are cocered ha he drbuo of he amle ac may be greaer ha he ull drbuo. Such he cae our fr examle. Defo 3: P VALUE m [ F (x OBS, F (x OBS ] Th defo ued whe we are cocered ha he drbuo of he amle ac may be eher greaer or le ha he ull drbuo. Noe ha h defo oly aled whe he ull drbuo ymmerc. H VALUE Fgure: Grahcal reeao of he ull drbuo. VALUE for defo 3 by area below le of dey of Whe he ull hyohe correc, he drbuo of he value uder all hree defo uform. Tha, P VALUE ( ( X < H Therefore, he value ha he ame erreao for all ull hyohee deede of he orgal ull drbuo. Clearly, maller value are more exreme wh reec o ull drbuo. Therefore, he maller he value, he roger he evdece of he amle ac aga he ull hyohe. Bu how mall mu he value be before he evdece rog eough o rejec he ull hyohe? Srcly eag, h would aga deed o he coex whch he hyohe eed. However, ce he wegh of evdece aga he ull hyohe creae cououly wh decreag value, would be ureaoable o degae a gle value cuoff o below whch he ull hyohe rejeced ad above whch acceed. Raher we hould exec a cocluve rego earag acce ad rejec value. 3

5. Cocluo erm of VALUE VALUE <, rejec H, < VALUE <,5 e cocluve VALUE >,5 acce H rejec H e cocluve acce H,,5 8.3. Alerae Hyohe From he defo of he value, clear ha he ure gfcace e rocedure for hyohe eg requre o oly a ecfc ull hyohe bu alo a oo of whch alerave mgh be correc f he ull hyohe rejeced. The alerae hyohe doe o eed o be ecfed a recely a he ull hyohe. To elec he arorae defo of he value, oly eceary o ow he dreco of he alerave wh reec o he ull. However, he alerae hyohe wll alo fluece he choce of he amle ac. Thoe value of he amle ac whch have a mall value uder he ull hyohe hould ed o have a larger value for roecve alerave ad vce vera. (Large ull value hould have mall alerae value. 8.4. Hyohe Te for Mea ad Meda Examle ad Soluo Coder he followg e IQ core. IQ e core are caled o have a mea of ad a adard devao of 5. 65 98 3 77 93 3 8 94 We wh o e he hyohe ha he mea. Soluo: We ca llurae h amle: 4

X 9.7 Meda 96 6 8 H : X (IQ ha N(, 5 ; m ; 5 Uder H Þ X N æ ç è, 5 ø ö X Z 9.7 ; X m 4.5 9.7 5.54 VALUE F(.54,678 Acce H : m Noe: a Whe he amle ze large, he Ceral Lm Theorem erm he ue of h e whe he orgal oulao o ormally drbued. b If o ow ad he orgal oulao o ormally drbued, he amle adard devao may be ubued whe he amle ze large. Examle ad Soluo Ue he ame daa a Examle Sude' e for mea of mall amle 5

X H : 9.7 ; 9 X je N(, ; m ; uow 4.5 X m 9.7. 59 4.5 VALUE (.59,7349 Acce H : m. Noe: a Whe he amle ze become larger ha abou 3, he drbuo become very mlar o he ormal drbuo. Examle ad Soluo 3 Ue he ame daa a Examle Sg e for meda A alerave o eg he hyohe ha he mea equal o e he meda. If he meda m, he he robably of ay obervao exceedg he meda.5. Therefore, he umber of obervao a amle of whch exceed he hyohezed meda wll have a bomal drbuo wh arameer ad.5. H : X (IQ ha meda m Le Y umber of obervao > m æ ö Y Bç, B(, è ø Y 4 value P( Y 4 4 æö ç è ø 386 4,377 The e reul how o coecy bewee he daa ad he hyohe. Noe: a The followg e mae o aumo abou he form of he orgal drbuo ad ca herefore be aled o ay drbuo. 6

b The g e ha lower ower ha he e whe he orgal drbuo ormal or he z e whe he Ceral Lm Theorem ale bu o affeced by dearure from hee codo ad o eve o ouler. Examle ad Soluo 4 Ue he ame daa a Examle Wlcoxo gedra e for meda A ecod alerave o e for amle mea baed o he ormal drbuo o relace he oberved value by her ra ad calculae a e ac from he ra. To e wheher he meda equal o ome hyohezed value m, we fr calculae he abolue dfferece of each obervao from m. The abolue dfferece are he relaced by her ra or he umber of her oo. The ra are he ged f he orgal obervao le ha m ad + f he orgal obervao greaer ha m. If he hyohe ha he rue meda equal m rue, he each ra or eger bewee ad, he amle ze ha equal robably of beg ove or egave. Therefore, he execed value of he mea of he ged ra hould be zero. Therefore calculag he mea ad adard devao of he ged ra ad formg he zcore a we do for he e would roduce a reaoable e ac. H : X (IQ ha meda m y x m r ra(y * r g ( x m r ged ra(y For he obervao of IQ core hee reul are a follow: IQ core Abolue Ra of abolue Sged ra r * dfferece y dfferece r 93 7 6 6 94 6 5 5 77 3 9 9 8 8 8 3 3 4 4 3 3 7 7 98 65 35 7

For he hree obervao whch have he ame abolue dfferece from he hyohezed meda, he average of he hree ra ha bee aged. The e ac of he ra calculaed a follow. Fr calculae he mea ad adard devao of he ged ra. r r *.5; r ( r r 5.9675 The calculae he zcore for he mea ged ra rememberg ha he execed value of he mea ged ra zero uder H. w r r.35 Acce H. VALUE W(.35 Φ (.35.957 Noe: a Le he g e, he Wlcoxo ged ra e mae o aumo abou he form of he orgal drbuo. If he orgal drbuo ormal, he Wlcoxo e wll have le ower ha he e, bu wll be le eve o dearure from he aumo of ormaly. A a geeral rule for mall amle reaoable o comue boh he uual e ad he Wlcoxo e. If he wo e gve very dffere value, h would ac a a warg ha he orgal drbuo may be erouly oormal. b Becaue he ra are fxed redeermed value, he Wlcoxo ac wll o be eve o ouler. c Comuaoally mler formula for comug he Wlcoxo e ac whch exlo he fac ha ra are fxed value are gve ome boo. 8.5. Error Through Teg Whe you mae a deco bewee comeg hyohee, here are wo oble of beg correc ad wo oble of mag a mae. Th ca be deced by he followg able. 8

Deco H True uao H A H OK Error II H A Error I OK If your deco lea oward H ad H deed rue (rue uao, he you dd o mae a error. If your deco lea oward H A ad H A rue, he aga you dd o mae a error. Thee are he robable aearg he uer lef ad lower rgh corer. The robable he lower lef ad uer rgh corer error made by o mag he correc deco. Thee robable are degaed by he Gree leer α ("alha" ad β ("bea". α called a Tye I Error. I he robably of falely rejecg H. I ofe referred o a he gfcace level ad reree he r oe wllg o ae rejecg falely. The uer or reearcher ha a comlee corol over α. Tycal (ad ubjecve α value are.5 ad.. If he coequece of a Tye I Error omehg he aure of creaed r of deah for a ae or creaed r facal loe, he oe would ue a level of gfcace o greaer ha.. β called a Tye II Error. I he robably of falely acceg H. Ule a Tye I Error, dffcul o quafy β. More wll be ad abou h laer. If he coequece of H A exremely aracve ad f he reul of a Tye I Error are o caarohc, may be advable o creae he r of mag a Tye I Error ad ue a level of gfcace ha.5 or hgher. Admedly dffcul a h me o fully comrehed hee coce. Hoefully hg wll mae more ee whe we go more deely o hyohe eg. P(Error I P(P VALUE < a H a P(Error II P(P VALUE > a H A b 9

8.6. Two Samle Te, Pared Samle Te ad Te for Prooro Examle ad Soluo 5 A uao whch are frequely racce he wo amle e. Two amle have bee obaed from dffere ource ad eceary o deerme wheher he wo ource have he ame mea or meda. Oe ource may be a corol grou ad he oher a exermeal grou. For examle, o deerme he effecvee of a ew eachg mehod, a corolled exerme may be coduced whch oe grou of ude, he corol grou, augh by radoal mehod ad a ecod grou by he exermeal mehod. The reearch queo h cae wheher he ude augh by he exermeal mehod aaed hgher reul. Samle from oulao # 6 49 5 68 68 45 57 5 3 4 33 3 8 3 48 Samle from oulao # 38 8 68 84 7 48 36 9 6 54 # Meda Mea # Mea Meda 4 6 8

The amle mea ad adard devao are: #: X 44.867; 5.77 # : Y 5.6; 7.93. We aume ha boh amle ue from ormal drbuo: X N( Le: 5 5 Te H : m m m, ;,..., Y j N(, ; j,..., The e ac X Y Z N(, + m, z 44.867 5.6 5 5 + 5.765 VALUE Φ(.765.. Sude' e for dfferece of mea The aumo of equaly of varace boh oulao reque he comuao of a gle emae of adard devao called he ooled amle adard devao. The ooled adard devao he average of quared devao of all obervao from he amle mea of her reecve oulao. If x he h amle obervao from oulao #, ad y j he j h amle obervao from oulao #, he he ooled adard devao ( x x + ( y y ( + ( + + j j Uder he aumo of equal varace boh oulao he emaed adard devao of he dfferece of amle mea wll be

x y + The he wo amle ac comued a: + x y x y x y + ad wll have a drbuo wh ( + degree of freedom. Alhough he aumo of equal varace queoable lgh of he dfferece amle adard devao, he wo amle e aled o he IQ daa above yeld he followg reul. 4 ( 5.77 + 9 ( 7.93. 37 3 44.867 5.6 3.37 + 5.77 VALUE (.77.4 + 3. Ma Whey or Wlcoxo ra e for dfferece of meda The wo amle ra e equvale o rag he oal amle from he wo oulao ad calculag he wo amle e ug he ra raher ha he orgal obervao. For he IQ daa, h gve he followg reul. Ra for oulao # 9 4 5.5 8 5.5 7 5.5 4 5.5.5 Ra for oulao # 9 3 4 3.5 8 5 7 The mea ad adard devao of he ra each oulao are

r r.; 4.35; r r 6.9 8.89 The ooled amle adard devao of ra The e ac r ( r + ( ( + 4 r ( 6.9 + 9 ( 8.89 7. 4 3 w r r r +.743 VALUE W (.743 Φ(.743.9 Pared amle e Whe eg he effec of ome exermeal codo or comarg he effec of wo dffere codo, he exermeal deg ofe ale eher codo or boh exermeal ad corol codo o he ame amlg u or h cae exermeal u. The raoale for h deg ha varao exermeal reul due o dfferece amlg u ca be elmaed leavg oly meaureme varao o obcure he effec of he exermeal codo. However, o ecure he beef of reduco varao offered by h deg, he arorae mehod of daa aaly ad coruco of e ac mu be aled. Suoe wo obervao uder dffere codo are ae of amlg u. For examle hear rae before ad afer exerce. Le X be he al oberved value for he h amlg u ad X he ubeque oberved value for he ame amlg u. Such a deg called a ared amle deg. I oble o aalyze h daa ad e he hyohe of o dfferece bewee he wo exermeal codo ug he wo amle mehod dcued above. However, h aroach would falure o ae advaage of he ooruy o elmae varao due o dfferece dvdual amlg u. A acally more effce mehod o aalyze h daa o ae advaage of he ared aure of he daa ad creae a gle value for each amlg u. I he mle daa model, h value would be he dfferece of he wo obervao for each amlg u. d X X The value d he reul oly of dfferece exermeal codo ad exermeal error. The mehod dcued he eco o oe amle e ca he be ued o e he 3

hyohee ha he mea or meda of d zero whch equvale o o dfferece bewee he wo exermeal codo. Examle ad Soluo 6 Coder he followg examle of he hear rae of ae a re ad afer e mue of exerce. Reg rae Rae afer Exerce Dfferece of Rae Sged Ra of 4 5 3.5 73 75 3 47 34 5 83 3 69 3 54 9 6 94 69 5 7 93 3 3 8.5 8 3 8.5 67 57 3.5 4 4 76 89 3 5 Dfferece The g e alo alcable o h daa. For h daa, here are 5 egave g ou of obervao. If he rue meda were, he umber of egave g ha a bomal drbuo wh arameer, ad.5 ad he robably f h eve : value 5 æö P Y 5 ç è ø (.387 For uch a mall amle wh uecfed oulao varace, we aume ha he obervao are ormally drbued ad aly he Sude' e. The mea ad adard devao of he ared dfferece are d d d 5 ( d d 6.86 4

ad he ac d 5 6.86 d.645 value from Defo (.645.66 Alyg he Wlcoxo ged ra e o hee daa yeld he followg reul. The mea ad adard devao of he ared dfferece are r r r.33 ( r r 7.55 ad he W ac r.33 W r 7.55.6 value Φ(.6.7 Te for rooro Whe eg hyohee abou he rooro of a oulao havg ome arbue, he amle ze,, wll be large eough mo cae o ue he ormal aroxmao o he drbuo of he amle rooro. Uder he ull hyohe ha he oulao rooro equal o ome ecfed value, H: The drbuo of he amle rooro wll be aroxmaely ormally drbued for large. æ ( ö ˆ N ç, ø è ad he value ca be calculaed from he zcore of he amle rooro. 5

æ ö ç ç ( ˆ value Φ ç ( ç è ø Examle ad Soluo 7 If he maufacurer' ecfcao for he defecve rae of a em o o exceed 3%, ad 7 defecve em are foud a amle of 95, he he value for eg he hyohe ha he amled oulao mee he maufacurer ecfcao 7 ( ˆ.3 z 95.5 ( (.3 (.97 95 value Φ(.5.6 Rejec H. Two amle e for rooro A wo amle e for rooro are whe amle are ae from wo oulao ad he ull hyohe o be eed ha he rooro boh oulao are he ame. If he amle from each oulao are large eough, he ormal aroxmao ca aga o aly o he drbuo of he dfferece of amle rooro. However, ce he ull hyohe doe o ecfy a gle value for each oulao, he varace emaed ug he oal rooro from he amle of boh oulao whch he maxmum lelhood emae of uder he ull hyohe of equal rooro boh oulao. Examle ad Soluo 8 Le X be he umber of em a amle of from oulao # havg he arbue ad X be he umber of em a amle of from oulao # havg he arbue. The ˆ X ; ˆ X ; ˆ X + X + The uder he ull hyohe ha he rooro he wo oulao are equal, H: 6

he drbuo of he dfferece amle rooro : ˆ ˆ æ N ç è, ˆ ( æ ˆ ç è ö + ø ö ø For examle, uoe defecve em were foud a amle of 88 from oe roduco ru ad 8 defecve em a amle 9 from a ecod ru. The, 8 ; ˆ ; ˆ 88 9 ˆ + 8 88 + 9 The he z ac for eg he hyohe ha he defecve rae boh ru he ame ˆ ˆ.36.87 z.54 æ ö æ ö ˆ ( ˆ (. (.99 ç + ç + è 88 9 ø è ø value Φ(.54. 46 Acce H. Examle ad Soluo 9 We have wo ye of floy d Soy ad 3M. I ay ace are d. There were foud 4 defecve d o 4 Soy ace ad here were foud 4 defecve d 3 3M ace. Doe dfferece he qualy of Soy ad 3M d ex? Soluo: ˆ ˆ 4.3 4. (rooro of defecve Soy d 4.3 3. (rooro of defecve 3M d ˆ 4 + 4.7 ( 4 + 3.. H : H A : >. We elec e ac 7

ˆ ˆ ( Z N (, æ ö ˆ ( ˆ ç + è ø ˆ ˆ (. 3. 3 æ ö ˆ ( ˆ. 7 (. 7 ( + ç + 8 6 è ø 8. 3. value F (P F (,8, F (,8,79 / (Z ha a adard ormal drbuo 4. value >>>.5 Þ acce H We ca' affrm ha here ex acal mora dfferece qualy of Soy ad 3M floy d. Summary The ure gfcace e awer he queo wheher gve radom amle X ( oberved value or o exreme relao o ome eed hyohe abou oulao. I co of 5 e. The la e coclude abou acceao or rejeco H. The hyohe e are mo ofe ued for mea ad meda: Sude' e, Wlcoxo e for meda. There ca be error cocluo of he ure gfcace e becaue o haeg a real uao. I cae of rejecg H whle rue, here he ye error. If H A acceed bu H hold he ye error. The followg e are he mo ofe ued: Sude' e for dfferece of mea, Wlcoxo ra e for dfferece of meda ad ared e. There are he mo ofe ued he e for rooro egeerg racce. Quz. How do you ge he P VALUE?. Wha he alerae hyohe? 3. Characerze wo amle e for rooro. 8

Praccal Exerce Exerce : Suoe you wa o how ha oly chldre have a average hgher choleerol level ha he aoal average. I ow ha he mea choleerol level for all Amerca 9. You e chldre oly ad fd ha he mea 98 ad he adard devao 5. Do you have evdece o ugge ha oly chldre have a average hgher choleerol level ha he aoal average? {Awer: rejecg H you ca coclude ha chldre do have a hgher average choleerol level he he aoal average. } Exerce : Ne dog ad e ca were eed o deerme f here a dfferece he average umber of day ha he amal ca urvve whou food. The dog averaged day wh a adard devao of day whle he ca averaged day wh a adard devao of 3 day. Wha ca be cocluded? {Awer: You fal o rejec he ull hyohe ad coclude ha here o uffce evdece o ugge ha here a dfferece bewee he mea arvao me for ca ad dog.} 9

9. POINT AND INTERVAL ESTIMATION Sudy me: 4 mue Learg Objecve you wll be able o Exla he roere of he o emao Coruc erval emao for mea, adard devao ad varace Exlaao 9.. Iroduco The emao roblem dguhed from hyohe eg. I hyohe eg we had a referece oward ull hyohe ad oly rejeced face of rog corary evdece. I he cae of emao, all arameer value or oeal hyohee are equal ad we wa o chooe a our emae hoe value whch are uored by or coe wh he daa. A emae by elf ju a umber. Ayoe ca mae a emae. To be uable, he accuracy of he emae mu alo be ow. Therefore addo o dervg emae, we mu alo mae ome aeme of he error of emao. 9.. Ierval Emao The objecve of erval emao o fd a erval of value whch have a hgh lelhood or robably of coag he rue arameer value. The raegy ued o fd hoe value whch would have a large value f hey had bee choe a he ull hyohe,. e. hoe arameer value whch are o coe wh he daa. I order o gve a robably erreao o he daa, we uually chooe a fxed value, eher oeded or woded deedg o wheher we wa a oe or wo ded erval, ad he clude our erval all arameer value whoe value for he oberved daa exceed he choe mmum value, α. The robably of he amle havg a value whch exceed he eleced value α, ad herefore he robably ha he erval o coruced wll clude he rue arameer value alo α. We call he value α he cofdece level of he erval. 3

Coder he examle of amlg emcoducor devce o deerme he rooro of defecve devce roduced. I h cae, uoe a ew roce ad we wh o emae a maxmum value for he rooro of defecve. A amle of devce eleced ad eed. Three are foud o be defecve. Sce we are ereed a uer boud, we a how large he rue rooro of defecve ca be before our oberved amle ha a very mall robably. For ome rooro,, of defecve devce, he robably of obag le ha 4 defecve a amle of Pr ob < x è x ø 3 æ ö x x [ X 4 ] ç (. To oba a (α uer for, we fd he value of uch ha he value exacly α 3 æö a ç x è x ø x x ( The followg dagram llurae he value a a fuco of he rooro of defecve, fd he value of whoe value α., ad defe he 9% uer cofdece lm for..9.8.7.6.5.4.3.. 9% Cofdece Ierval Prob[ X < 4 ]...3.4.5.6.7.8.9 The robably ha a oulao wh 47% defecve wll have le ha 4 defecve a amle of %. Thu, we are 9% cofde ha he rue rooro of defecve le ha 47%. Aoher way of exreg h dea o ay ha 9% of cofdece erval calculaed by h mehodology wll clude he rue rooro of defecve. Coder a ecod examle. A frm whch aemble PC' from bac comoe, load he ofware, ad e he yem before delvery ereed emaed how log ae a worer o comlee rearao of a PC for delvery. They oberve a worer for 4 3

hour, oe half of h daly wor erod. I ha me, he worer comlee 7 PC'. If we aume ha he me o comlee a gle PC exoeally drbued he he umber of PC' comleed 4 hour wll have a Poo drbuo. The frm ereed emag a uer boud for he mea me o comlee a PC or equvalely a lower boud for he rae a whch PC' are comleed. Therefore we a how low he rae λ ca be before he robably of our amle reul, more ha 6, ha a very mall robably. For a gve value of λ, he value of our amle Pr ob [ ] X > 6 x x l l e l. 7 x! The followg dagram llurae he value a a fuco of he comleo rae λ, fd he value of whoe value α., ad defe he 9% uer cofdece lm for λ..9.8.7.6.5.4.3.. Prob[ X > 6 l ] 9% Cofdece Ierval 3 6 9 5 If he rae of rearg PC' for delvery er 4 hour erod, he he robably of comleg 7 or more PC' 4 hour %. Therefore, we are 9% cofde ha he rue rae of rearao a lea 3.9 comuer er 4 hour, or lghly le ha oe comuer er hour. Aleravely he mea me o comlee each PC o more ha 6 mue 3.3 ecod. Th obvouly a coervave emae ce our amle, comuer were comleed a he rae of 7 er 4 hour or.75 er hour wh a equvale mea rearao me of 34 mue 7 ecod. To oba a le coervave emae a he ame cofdece level, a larger amle ze requred. Th aaly deed o he aumo ha he me o comlee a PC exoeally drbued. I racce h ulely o be a very good model becaue heory accordg o he exoeal drbuo, he PC could be comleed aaeouly. The Poo roce a more arorae model for eve whch occur radomly uch a raffc accde. 3

Now coder a examle where we wh o emae boh a uer ad lower boud for he arameer. I h cae, we ue he value for eg hyohee aga woded alerave. The α cofdece erval he e of all arameer value havg a value greaer ha α. Fle ramed va comuer ewor are ofe budled o grou of fle havg mlar ewor ahway. Io order o deerme he omal umber of fle o clude a gle budle, ewor egeer eed ome emae of he drbuo of fle ze. A amle of 5 fle ae wh he followg reul. Sze are MB u 4.7.887 3.86 7.8.753 5.956 8.7.857 4.55 7.8.4 6.86 5.7.558 5.534 The ummary ac for hee daa are Mea 5.55 Sadard. Devao..646 Meda 5.7.483*MAD.739 Shorh 3.86 o 7.8 For ay hyohecal value of he rue mea fle ze, we ca comue he ac for our oberved amle. ( m x m ad aocaed value. Sce h cae, we wa boh uer ad lower boud for our emae of µ, we ue he woded defo of value. ( x m m{ F [ ( x m ], F [ ( x m ]} For examle for a hyohecal value of µ 6, he oberved value ad he aocaed value 5.55 6.646 5 4.384 ( x Pr ob[ 4 ³.384]. 88 m 33

.4.3. X 5.55 value.88. m 6 Thu, we ca deerme he value aocaed wh ay value of µ. The (α cofdece erval wll co of all value of whoe value greaer ha α. The followg char how he woded value a dffere value of µ. The value reache maxmum value of whe µ 5.55, he amle mea. If we clude our erval emae all value of µ havg a value of a lea %, he we cae be 9% cofde ha he erval emae coa he rue mea value he ee ha 9% of erval emae derved by h rocedure coa he rue value of he mea. We call uch a erval a 9% cofdece erval. I h examle, he 9% cofdece erval for mea fle ze Two Sded value for Samle Mea 5.55.75.5.5 a. 9% Cofdece Ierval 3 4 5 6 7 8 he rage (3.85, 6.58. Noce ha µ 6 wh a value of.88 cluded he 9% cofdece erval. 34

9.3. Coruco of Cofdece Ierval There a mle rocedure for corucg cofdece erval for arameer whoe e ac ha a ymmerc drbuo, uch a he Sude' or he ormal. Th rocedure elmae he eed o comue he value for every value of µ. To coruc a (α cofdece erval, we eed oly deerme he uer ad lower boud of he erval. The lower boud wll be ha value of µ le ha he amle mea whoe value exacly α. Therefore he value of he amle mea wh reec o he lower boud mu be equal o he (α / erceage o of he Sude' drbuo., a x m Lower Solvg for µ Lower, we have m Lower x a, Smlarly µ Uer mu afy he equao, a m Uer x Hece he ( α cofdece erval gve by ( m m Lower, Uer Û x ±, a I our examle of fle ze, he 9% cofdece erval.646 x ± 4,.5 Û 5.55 ±.76 Û 5.55 ±.3 5 A revouly deermed grahcally, h erval (3.85, 6.58. 9.4. Samle Sze Deermao A well a gvg a rage of reaoably good arameer value, a erval emae alo rovde formao abou he qualy of he emaed value. The qualy of a emae ha wo aec, Accuracy Relably 35

The accuracy of a erval emae equvale o he legh of he erval. The maller he erval, he greaer he accuracy. Relably gve by he cofdece level of he erval. However, a for Tye I ad Tye II error of hyohe e, accuracy ad relably of a cofdece erval are coflc. For a fxed amle ze, he cofdece level ca oly be creaed by creag he legh of he erval hereby reducg accuracy. Icreag boh he accuracy ad relably ca oly be acheved by creag he amle ze. Deermg he amle ze requred corucg a erval emae havg ome fxed relably ad accuracy a roblem whch are commoly racce. Suoe eceary o emae a mea o a accuracy of Δ wh a relably of (α. Tha, we requre a (α cofdece erval of legh o exceedg Δ. The he amle ze mu be choe large eough o afy he followg equaly. D ³, a ad herefore he amle ze mu exceed é ³ ê ê ë, a D ù ú ú û I racce commo o ubue a coervave emae for ad ubue he z quale for he quale o he aumo ha he requred amle ze wll be large eough ha he alcable drbuo wll be aroxmaely ormal. If we whed o emae fle ze o a accuracy of 56 KB, or.5 MB wh 9% cofdece, ug a coervave emae of 3 for, we would requre a amle ze of é z ³ ê ê D ë a ù ú ú û é.645*3ù ê.5 ú ë û 389.67 9.5. Po Emao Radome dffcul ad uoular. We are ued o have ecfc awer o queo. A erval of emae ca be uafyg. We may a "Wha he gle be o or value he erval?" Such a gle value would be a o erval. The gle be value clearly he value whch ha he hghe value for he oberved daa. Th o emae called he maxmum lelhood emae. Bu oce ha a he value creae ad he ze of he cofdece erval decreae, he cofdece level decreae accordgly. I mo cae, he maxmum value wll be, o he cofdece level wll be zero. Tha, he o emae wll ever be exacly correc. Therefore h cae exremely mora o emae he error of emao. Whle choog he gle o emae o be he arameer value agg he maxmum value o he oberved daa a logcal coequece of he mehod of corucg 36

cofdece erval, may have deermae or ouque oluo cera cae, arcularly for dcree radom varable. Therefore, o emae are deermed by a mehod mlar r o maxmzg value, he mehod of maxmum lelhood. Raher ha maxmzg he value, maxmum lelhood o emae ee he arameer value whch maxmze he robably ma or robably dey of he oberved amle daa. 9.6. Maxmum Lelhood Emao Becaue ac meaure geeral drbuoal roere uch a locao ad cale, mea, meda, ad adard devao ca be aled o ay drbuo. Bu emaor are aocaed wh arameer for a ecfc drbuo. Develog afacory emaor for every dvdual drbuoal form would become a daug a whou ome geeral rocedure or aroach. Foruaely, he dea of lelhood offer uch a aroach. Iuvely, f he codoal robably or lelhood of he oberved daa greaer for oe arameer value ha aoher, he he fr arameer value a referred emae of he oulao arameer. By exeo, he be choce of emae for he oulao arameer hould be he arameer value whoe lelhood maxmum. qˆ max Pr ob q [ x q ] max f ( x, q q A emaor derved by h crero called a maxmum lelhood emaor or MLE ad alway deoed wh a mall ca over he arameer ymbol a dcaed. Coder he cae of amlg for a arbue. If X he umber of em a amle of ze havg he dered arbue, he X wll have a bomal drbuo wh arameer whch ow from he amlg rocedure ad whch uow. The lelhood of for he oberved X he codoal robably of X gve. f ( x æö ç è xø x x Pr ob [ x ] ( We ca fd he maxmum lelhood emae of by fdg he o of he lelhood fuco havg zero loe. Tha, by olvg he followg equao. f ( x æö x ç ( è xø æö ç è xø x x x x x [ x ( ( x ( ] The oluo he amle rooro. We ow from he amlg drbuo of h emaor ha ubaed. I alo coe, uffce, ad effce amog ubaed emaor. 37

ˆ The maxmum lelhood emae for ad X 7 lluraed below. Maxmum Lelhood.3 Emae of whe X 7.5..5..5 x...3.4.5.6.7.8.9 The maxmum lelhood emaor of λ for he Poo drbuo derved he ame faho. f ( l x Pr ob [ x l] ( l x l e x! Solvg for he value of λ where he loe of he lelhood fuco zero f ( l x l x ( l x e x! l l x l ( l e ( l x! x e l we fd ˆ x l. The maxmum lelhood emae of λ whe X 3 ad lluraed below. 38

.5. Maxmum Lelhood Emae of l whe X 3.5..5 3 4 5 6 7 8 9 The maxmum lelhood emaor of λ alo ubaed, effce, coe, ad uffce for he Poo drbuo. For he ormal drbuo, he maxmum lelhood emae of µ obaed by mmzg he value he exoe of he dey. m m ( x m The oluo he amle mea. 9.7. Emao, Emaor, Proere of Emaor Wha a Emae or a Emaor? Formally, a emaor a ac defed o he doma of he amle daa whch ued a a emae a oulao arameer. A emae he value of ha ac for a arcular amle reul. Sce every ac ha a heorecal robably drbuo for every hyohecal robably drbuo of he oulao, oble o exame he roere of a emaor hrough robably drbuo. Sce may emaor are baed o lelhood fuco, may of he roere of he lelhood fuco wll alo be exhbed he robably drbuo. Becaue of he ecal requreme of a emaor, crera arcular o he roblem of emao have bee rooed a mea of evaluag he uably of a ac a a emaor, of comarg comeg emaor, ad of develog ew ad mroved emaor. You wll o ha he followg crera ca oly be aled o ac whch are requred o be cloe o ome arameer, hece o emaor. a Coecy 39

Coecy geerally agreed o be a eeal characerc of a emaor. A he amle ze creae, a emaor whch coe wll have maller ad maller robably of devag a ecfed dace from he arameer beg emaed. I he lm of a fely large amle, he value of he emaor wll be equal o he emaed arameer wh robably oe. A emaor whch doe o have h roery would gve more relable reul for maller amle ad herefore could o be ug he formao he amle coely. b Suffcecy Every ac a reduco or ummarzao of he orgal amle daa ad a uch dcard ome formao coaed he orgal amle daa. A emaor whch uffce doe o dcard ay formao releva o he arameer beg emaed. Th may eem a fr o be a raher vague requreme bu fac uffcecy ha a very ecfc robablc defo. Ay ac creae a aro of he amle ace. All eleme wh ay aro lead o he ame value of he ac. If he relave or codoal robable of he dvdual amle ace eleme are deede of he arameer beg emaed, he he aro ad he emaor whch creaed are uffce. For examle, he bomal radom varable aro he amle ace of Beroull ral o ube all havg he ame umber of uccee. The robably of wo eleme havg he ame umber of uccee alway he ame o maer wha he value of, he robably of ucce. Therefore o furher formao abou ca be obaed by owg whch eleme he aro acually occurred. Therefore, he amle rooro a uffce ac or emaor of. We alway ry o wor wh uffce ac. c Ba Ba or ubaede cocer he execed or mea value of he ac. The ac hould be cloe o he arameer beg emaed ad herefore mea value hould be ear he arameer value. Ba he dfferece bewee he mea of he emaor ad he value of he arameer. Ba hould be mall. If ba zero, we ay he emaor ubaed. Ubaed emaor are o alway referable o baed oe f hey are o uffce or have larger varace. A emaor eed o be ubaed order o be coe. d Effcecy A ha be ad everal me ad reforced by he oreao of he recedg crera, a emaor hould be cloe o he arameer beg emaed. Smly beg ubaed wll o ure ha he emaor cloe o he arameer. The varace of he amlg drbuo of he emaor mu be mall a well. For wo emaor, he emaor whoe um of varace ad quared ba maller ad o be more effce. If he wo emaor are ubaed, he he emaor wh maller varace more effce. If a ubaed emaor ha mmum oble varace for all ubaed emaor, he ad o be effce. 4

Summary From a mehodologcal o of vew we ue wo d of arameer emao. I a o emao where drbuo arameer aroxmaed by a umber ad o called erval emao where h arameer aroxmaed by a erval where he arameer belog wh a hgh robably. Ubaed, coe ad effce emao of arameer are he mo mora for a qualy of o emao. I he cae of erval emao of a arameer we ca earch for woded or oeded emao. Quz. Wha he coe emao of arameer?. How ca you decrbe.(a % woded cofdece erval for a θarameer? Praccal Exerce Exerce : I a radom amle of che % o uable for ew qualy demad. Fd 95% cofdece erval for a che rooro o uable for a ew orm f a amle ze : a b 5 c 5 d { Awer: a.6<<.6, b.<<., c.3<<.7, d.7<<.3} Exerce : Four ude were radomly eleced from he fr arallel grou. Ther exam reul were 64, 66, 89 ad 77 o. Three ude were radomly eleced from he ecod arallel grou. Ther exam reul were 56, 7 ad 53 o. Fd 95% cofdece erval for dfferece bewee mea value of exam reul. { Awer: (4;3} 4

. ANOVA Oe Facor Aaly of Varace Sudy me: 6 mue Learg Objecve you wll be able o Exla he rucure of Frao Coclude by he e amed Aaly of Varace Coruc he ANOVA able Carry ou he o hoc aaly Exlaao.. Iroduco We dcued oeamle ad woamle mea e he revou chaer. The aaly of varace (ANOVA a exeo of hoe e. I eable o comare ay mea of deede radom amle. The aaly of varace ( aramerc form aume ormaly of he drbuo ad homocedacy (decal varace. If hee codo are o execued he we mu ue oaramerc KruallWall e. I a aalogy of he oefacor org he aaly of he varace. I doe' aume drbuo ormaly bu dadvaage maller evy... Coruco of he FSac Le' have radom amle ha are deede o oe aoher. Thee amle have he adard drbuo wh he ame varao: (X (X... (X, X, X, X N,..., X,..., X,..., X N( m, N( m, N( m,, Le... umber of obervao h amle. 4

Formulao of he roblem: The hyohe of ere H : m m... m m The alerae hyohe : H A : A lea wo m ' are dffere. We wa o deerme H oe e becaue we ry o fd a e ac ha eable o oly H mlemeao bu alo eve o he H valdy. Defe he oal um of quare (or oal varably a SS TOTAL j ( X X, where X he mea of all obervao. j he oal um of quare he raw meaure of varably he daa Th oal um of quare ha comoe: SS TOTAL j ( X X Þ SS SS + SS, j TOTAL W where SS W... he wh grou varao (he um of quare wh grou he raw varably wh amle SS W j ( X j X B ( he degree of freedom equal o he um of he dvdual degree of freedom for each amle. Sce each amle ha degree of freedom equal o oe le her amle ze, ad here are amle, he oal degree of freedom le ha he oal amle ze: N S a amle adard devao of h radom amle: ad S ( X j X j X j j X a he amle mea he h amle. SS B... he bewee grou varao (he um of quare bewee grou he raw varably bewee amle: SS B ( X X SSW The wh grou varace (mea quare wh grou: SW N SS B The bewee grou varace (mea quare bewee grou: S B Proere of hee varace: S 43

44. ( ( ( ( W S E N S E N ES becaue ( S E. The wh mea quare a ubaed emae of he varace, deede of H.. + B X E E X ES ( Û B ES whe H rue Therefore he rao of he wo um of quared varace dvded by her degree of freedom wll have a F drbuo uder he hyohe of equal oulao mea. W B W B S S N SS SS F Defo: We call h F ac a Frao. Why ueful ue Frao a he e ac? We ee ha f H rue he Frao ay radom umber cloe o...» F. If H fale he h umber maredly bgger ha (ee roery. The ac Frao eve o valdy of he hyohe H. So we ca ue durg he followg eg a e ac ad we have o deerme acal behavor whch mea o deerme roably drbuo. We ow ha ( ( ( N S N S w c, becaue ( ( S c, ad furher ow ha he um of radom varable ( c aga a radom varable of he ame ye wh degree of freedom umber ame a ummarzed varable. If H rue he: ( ( ( ( X X X X S B c The we ow (FherSedecor drbuo ha he followg rao: N W B w B F S S N N S S, ( (

mu have Fdrbuo wh ( ad (N degree of freedom. If we ow a Frao acal behavor we ca ue for aaly ad deermao of revouly aed roblem H. Followg fgure llurae how Frao ued o deerme a hyohe H valdy. F Area of H valdy P value oberved value of he Frao ac.3. ANOVA Table We ummarze he daa a ANOVA able: Source Sum of quare Degree of freedom oal SS TOTAL j ( X j X N Mea quare Frao Pvalue bewee wh SS SS B W ( X X j ( X X N j S S B W SS B SSW N F S S B W ee defo Aaly of varace able ANOVA The bg value of Frao dcae mall value of value whch mea rejeco of H. The Frao value wll be a bg umber f he wh grou varao a eglgble ar of he oal varably ad equvalely f he bewee varao a gfca ar of he oal varably. 45

.4. Examle ad Soluo We aume hree daa e for llurao of Frao acal behavor. I each daa e, he amle mea are he ame bu he varao wh grou dffer. Whe he wh grou varao mall he he he Frao large. Whe he wh grou varao large he he Frao mall. The examle llurae hree cae: a mall wh grou varao; a ormal wh grou varao; ad a large wh grou varao. Examle : Small wh grou varao Grou I II III IV 4 7.5 68.5 38 34.5 7 44 3.5 6 53 5 Daa 4 5 64 5 46.5.5 57 43.5 8 3 56 4 37 5 54.5 4 35.5 6.5 46 63.5 37.5 6 36 66 55 Samle ze 8 7 Grou mea 37 7 6 43 Grou adard devao 5.78 3.7 6.6 5.7 Pvalue. ANOVA Table Degree of Sum of Mea quare Frao freedom quare oal 36 987.77 bewee 3 89.77 967.57.96 wh 33 97 9.39 46

8 7 6 5 4 3 I II III IV Examle : Normal wh grou varao Grou I II III IV 47 8 76 33 3 7 83 45 8 5 45 6 Daa 43 3 67 57 56 4 53 44 9 9 5 39 37 3 48 4 34 64 49 66 3 59 9 7 49 Samle ze 8 7 Grou mea 37 7 6 43 Grou adard devao.56 7.4..53 Pvalue. ANOVA able Degree of Sum of Mea quare Frao freedom quare oal 36 78.77 bewee 3 89.77 967.57 5.4 wh 33 388 7.58 47

9 8 7 6 5 4 3 I II III IV Examle 3: Large wh grou varao Grou I II III IV 67 6 3 3 7 49 3 97 Daa 55 5 79 85 94 38 37 46 7 53 3 3 37 5 37 8 7 6 76 55 9 5 Samle ze 8 7 Grou mea 37 7 6 43 Grou adard devao 34.69.5 36.36 3.59 Pvalue.549 ANOVA able Degree of Sum of Mea quare Frao freedom quare oal 36 438.77 bewee 3 89.77 967.57.84 wh 33 349 58.8 48

4 8 6 4 I II III IV.5. Po Hoc Aaly A large Frao dcae oly ha ome dfferece ex amog he grou mea, bu o where hoe dfferece occur. If he Frao large, our aaly would be comlee whou defyg whch grou mea dffer. Th roce called o hoc aaly, ad co of comarg he mea of all ar of amle o deerme f here a dfferece of mea. Several mehod are avalable for o hoc mulle comaro. We wll dcu he mle mehod here called Lea Sgfca Dfferece. The Lea Sgfca Dfferece or LSDmehod co of alyg he woamle e o every ar of amle mea. However, here oe adjume he quare roo of mea quare wh ued raher ha he ooled adard devao from he wo amle a he emae of oulao adard devao. Thu for ay ar of amle mea, he LSD comued a: ( LSD, j j N S W X X + j where S W S W SSW N. Th ac ha he Sude drbuo wh N degree of freedom. The LSD mehod lluraed for he hree examle gve revouly. Examle : Small wh grou varao We deerme (LSD,j for all ar of gve four grou ad he obaed value recorded he followg able: 49

Samle 8 7 ze I II III IV 8 I 7.8 9.698.333 7 II 7.8 7.64 9.73 III 9.698 7.64 7.754 IV.333 9.73 7.754 I h cae, here a very rog evdece of dfferece bewee all grou exce I ad IV where he evdece weaer. Examle : Normal wh grou varao Samle 8 7 ze I II III IV 8 I 3.564 4.849.67 7 II 3.564 8.53 4.8656 III 4.849 8.53 3.877 IV.67 4.866 3.877 wo homogeou grou: I ad IV I h cae, eve hough he amle mea are he ame, here o evdece of dfferece bewee he mea of Grou I ad IV. Therefore here are eeally hree Grou: II; III; ad I ad IV ogeher. I II III IV Samle mea 5

Examle 3: Large wh grou varao Sce he Frao for h examle very mall, we would ormally coclude ha here o evdece aga he ull hyohe of equal grou mea ad o roceed furher. Ay woamle e whch roduce mall value hould be regarded a urou. However, for llurao, le have a loo a he able of lea gfca dfferece. Samle 8 7 ze I II III IV 8 I.88.66.389 7 II.88.844.6 III.66.844.9 IV.389.6.9 I h examle, he oly lea gfca dfferece whch ha a mall value bewee grou II ad III. However, becaue he overall Frao wa oo mall, h dfferece would be dregarded wh cocluo ha o dfferece ex bewee he mea of ay of he grou. Noe: Aar from he LSD mehod here are alo oher e ha offer mlar mulle comaro le he o hoc aaly. More flexble mehod were alo develoed ad are acceble hrough he more advaced ofware (e.g. Duca e, Tuey e for gfca dfferece, Scheffe e ad Bofero e. Thee e are baed o mlar deco raegy ad ha' o eg of a crcal dfferece requeed for deermao f wo amle mea from everal grou are dffere. I may cae hee e are much more effecve ha he LSD mehod..6. KrualWall Te The Frao e ac ued he adard aaly of varace ow o be very eve o he aumo ha he orgal obervao are ormally drbued. Becaue he e ac baed o quared devao from he mea, ca be badly dored by ouler. For woamle aaly, he Wlcoxo/Ma Whey ra e wa roduced a a oaramerc alerave whch le eve o ouler ha he e. For mulle amle, he KrualWall ra e ca be ued for he ame uroe. Le he Wlcoxo/Ma Whey e, he KrualWall e ubue he ra of he orgal daa value ad erform a aaly of varace o he ra. For he large devao daa of he revou examle he ra for each grou are led he followg able. 5

Grou I II III IV 8 36 9.5 Ra.5 37 3 of orgal 6.5 8 9.5 35 daa 5.5 4.5 3 3 34 9 4 6.5 6.5 9 4.5.5 9 5 9 7 3 6.5 5.5 3 33 4 Samle ze 8 7 Mea ra 7.6875.743 4.467 9.35 Sadard devao.674 8.599 9.6668.6538 The e ac a modfcao of calculag he Frao for he ra. I h examle, he e ac ad value are: KW e ac 7.435 value.645 The value for he KrualWall e lghly hgher ha for he Fe, bu he cocluo are he ame boh cae. The ull hyohe of equal grou mea o rejeced. 5

Summary Aaly of varace (ANOVA a exeo of he woamle e for mea ad eable o comare ay mea of deede radom amle. Frao he e ac aaly of varace. Frao ac eve o valdy of he hyohe H, whch formulaed a a equaly of he amle mea. Parcular erreul (ha we execue durg aaly of varace are recorded o ANOVA able. The ecod e ( ANOVA o hoc aaly, ad co of comarg he mea of all ar of grou of uroe o chooe homogeou grou. LSDac a crero for agme o homogeou grou. Decrbed rocedure ANOVA eve o he aumo ha he orgal obervao are ormally drbued. If h codo o execued he we mu ue oaramerc KruallWall ra e. Quz. Decrbe coruco ad acal behavor of he Frao ac.. Wha he uual ouu from he aaly of varace? 3. Wha he o hoc aaly? 53

Praccal Exerce Exerce : A reearch aeg deedecy of earg o acheved educao ha bee carred ou. I he able here are earg houad of CZK of 7 radomly eleced me a each level of educao. (B bac, H hgh, U uvery. {Awer: value.57} B H U.9 8.9. 9.8.3 9.7 3 6.4 7.5 5.8 4 4.3 6.9 8.9 5 7.5 4.. 6.3 9.3 7.5 7 5..5. Do a mle org ad deerme f educao fluece earg. Exerce : From a large e of home you radomly eleced 5 gleoccua home, 8 home occued by coule, home wh hree famly member, home wh four famly member ad 7 home wh fve famly member. The her mohly edg for food ad dr er oe famly member ( CZK wa recorded. Cofrm by aaly of varace f a mohly edg for food ad dr deed o a umber of famly member. Number of famly member Sedg for oe famly member ( CZK 3 4 5 3.44.35.59.37.6 4.44 3.3.35..39 4.4.43.73.786.448 3.776.36.33.3.37 3.67.8.33.3.3.9.565.433..656.777.4..878.899.763.755.3 3.54.66 {Awer: Ue uable ofware acage.} 54

. SIMPLE LINEAR REGRESSION Sudy me: 6 mue Learg Objecve you wll be able o Exla a geeral lear model Exla a lear regreo model rcle Ue regreo aaly reul Verfy a regreo model by deermao dex Exlaao.. Iroduco Mahemacal formulao of acal model Symbolcally, he bac addve formulao of acal model ca be exreed a Y f X ( + z ( e where Y he oberved value, f(x he yemac comoe ad ζ(ε he radom comoe. Th chemac model exlcly defe hree ye of varable. Y Reoe, Crero, deede Varable (oberved value of rmary ere X Predcor, Smulu, Ideede Varable (hoe facor o whch he value of he yemac comoe may be arbued ε radom error Oly Y ad X are obervable. Radom error alway uobervable. ζ(ε alway emaed a he redua dfferece bewee he emaed yemac comoe ad he oberved reoe, Y. z ( e Y f ( X 55

Therefore he emaed l of he oberved reoe o yemac ad radom comoe a much a coequece of he choce of model, f ad ζ, ad he mehod of emao a of he oberved mulu ad reoe, X ad Y... Geeral Lear Model The geeral lear acal model a ecal mle cae of he chemac acal model dcued above. The ocalled lear acal model ulae ha he yemac comoe a lear combao of he yemac facor or varable, ad he radom comoe he dey fuco of radom error. radom comoe: z ( e e f X b yemac comoe: ( X b + Why ue a Lear Syemac Fuco? Lear yemac comoe have hree fudameal roere whch are derable for acal model mlcy, emably ad ably. Lear fuco reree or gve algebrac exreo o he mle d of relaoh. Lear fuco oulae eher: mulu ad reoe ed o creae ad decreae ogeher reoe decreae a mulu creae For he mle lear model Y b + b X + e f β <; he relao egave > Y decreae a X creae f β >; he relao ove > Y ad X creae ogeher Aumo abou he radom comoe I decreag order of mac o reul ad erreao, he followg hree aumo abou he behavor of he radom comoe of a lear acal model are wdely adoed.. Ideedece he radom error ε ad ε j are deede for all ar of obervao ad j. Equal Varace he radom error ε all have he ame varace σ for all obervao 3. Normaly he radom error ε are ormally drbued 56

.3. Emao of Parameer for he Smle Lear Regreo Model The followg caer lo llurae he ye of daa whch ycally decrbed by a mle lear model. 8 6 4 4 6 8 From he formulao of he geeral lear model, he ecal cae of he mle lear model whch he yemac comoe a lear fuco of a gle varable, ha a ragh le, may be exreed a: ad all ε are muually deede. Y b + b X e N (, For ay emae of he arameer, β ad β, ay b ad b, he redual error of emao are: + e e Y b b X a lluraed below. 57

58 The lea quare arameer emae are hoe value of b ad b whch mmze he um of quared redual error. ( ( x b b y e b b S, To fd he arameer emae whch mmze he um of quared redual, we comue he dervave wh reec o b ad b ad equae hem o zero. ( ( ( (,, e x x b b y x b b b S e x b b y b b b S The oluo o he above equao are he lea quare arameer emae. Noce ha he fr equao ure ha he redual for he lea quare emae of β ad β alway um o zero. Becaue he lea quare emae are alo maxmum lelhood emae uder he aumo of ormally drbued error, hey are uually deoed by he ymbol β ad β. The oluo o he lea quare equao are: ( ( ( x y xy r x x y y x x x y ˆ ˆ ˆ b b b The erce arameer, ˆb, merely lace he vercal oo of he le a he o where he redual error um o zero. The oerave arameer he loe emae, ˆ b, whch ha a 4 6 8 4 6 8 e

arcularly mle form erm of he correlao ad relave adard devao of he reoe Y ad he exlaaory varable X. ˆ b Relao Bewee X Scale Facor } } ad Y y r xy x The redual um of quare for he mle regreo model S ( bˆ, bˆ ( y bˆ bˆ x ( r ( y y ( r ( xy xy whch le he lea quare loe emae, b ˆ, ha a mle exreo erm of he correlao bewee X ad Y ad he varace of Y. The redual um of quare for a regreo model meaure how well he model f he daa. A maller redual um of quare dcae a beer f. Becaue a hgher quared correlao bewee X ad Y aocaed wh a maller redual um of quare a a rooro of he varace of Y, he quared correlao bewee X ad Y uually ued a a meaure of he goode of f of he regreo model. Whe r xy ±, he amle obervao of X ad Y all le o a ragh le ad he redual um of quare zero. Whe r xy, X ad Y are deede ad he redual um of quare wll equal he um of quared devao of Y abou mea. If he redual um of quare meaure he ze of he radom comoe of he regreo model, he he remader, he dfferece bewee he orgal um of quared devao of Y abou mea ad he redual um of quare of Y abou he regreo le mu reree he yemac comoe of he model. To beer uderad wha h yemac comoe meaure, le he o o he regreo le or redced value of Y for he h obervao of X be ˆ ˆ ˆ. y b + bx Frly, oe ha he lea quare emae of ˆb ad b ˆ ure ha he mea of he redced value of Y wll alway equal he mea of he orgal obervao of Y. Tha, y yˆ y ˆ b ( x x y. The a wa he cae he aaly of varace, he oal um of quared devao of Y from mea SS Toal ( y y 59

ca be aroed o he um of quared redual error, SS Error ( y b ˆ ˆ b x ( y y ˆ ad he um of quared devao of he redced value of Y from her mea. We ee ha SS Toal SS Re greo SS ˆ Re greo ( y y ( y y ( y yˆ + yˆ y ( yˆ y + ( y yˆ + SS Error The um of quare due o regreo ofe called he exlaed varao ad coverely he um of quared redual error, he uexlaed varao. The arog of he oal varao of Y o hee wo comoe due o he fac ha he lea quare emae mu afy he codo, x e Tha, he redual error mu be orhogoal o he redcor varable. The arog of he oal um of quared devao of he reoe, Y, abou mea o he yemac comoe, exlaed varao, ad he radom comoe, um of quared redual frequely reeed a a Aaly of Varace able. The Fe comued by h ANOVA Table e he ull hyohe ha he yemac comoe of he model zero. Source Degree of Freedom Sum of Square Mea Square Frao y Toal ( Regreo Error r ( r ( xy y xy y ( ( ( r r xy ( xy y ( y ( r xy ( r Thu, he Fe for eg he gfcace of he regreo model deed oly o he correlao bewee reoe ad exlaaory varable ad o he amle ze. I racce, he ull hyohe of o regreo effec almo alway rejeced, bu eve f rejeced doe o mly ha he regreo model wll rovde afacory redco. xy 6

A he cae of aaly of varace for facoral model, he emae of he error varace,, he mea quared error. ˆ SS Error æ ö ( r ç xy è ø y Th emaed error varace for he regreo le alo called he codoal varace of Y gve X, ha, he varace of Y remag afer he effec of X ha bee removed. y x ˆ æ ö ( r ç ø xy è y A ecod coequece of lea quare emae of b ad b ha he lea quare le wll alway a hrough he o of mea ( X, Y. I fac he zvalue of he redco for Y mly he correlao bewee X ad Y me he correodg zvalue for X. Tha, yˆ æ y çr è y x ö æ x + çr ø è y x ö x ø æ y ( yˆ y çr ( x x ç x è æ ö æ ö ç ŷ y x x. r ç è y ø è x ø ö ø Clearly whe x x, he ˆ y. y.4. Drbuo of Lea Square Parameer Emae If he redcor or exlaaory varable X aumed o be a fxed coa raher ha a radom varable, he boh ˆb ad b ˆ are lear combao of he ormally drbued crero or reoe varable, Y, ad hece are ormally drbued hemelve. The mea ad varace of he loe arameer emae are E V [ bˆ ] [ ˆ b ] b ( x x ( Thee reul ca readly be eablhed by og ha he lea quare emae of b may be exreed a he followg lear combao of he obervao of Y. x 6

ˆ b ( x x( y y ( x x ( x x y ( x The he execed value of he loe arameer emae E ( ˆ ( x x b ( ad he varace of he loe emae V E ( y ( x x( b + bx ( x ( ˆ ( x x b ( V ( y ( x x ( x x 4 x. The gfcace of hee reul ha he lea quare emae of he loe arameer ubaed ad varace become maller a he amle ze creae. I addo, he varace of he emae become maller whe he varace or rage of X become larger. By ubuo of he mea quared error emae of o he exreo of he varace of he loe arameer emae, he followg emaed varace of he loe arameer obaed ˆ ˆ b SS Error ( ( x ( r xy ( Becaue b ˆ ubaed, ubuo o he lea quare deermg equao for ˆb readly how ha he lea quare erce emae, ˆb, alo ubaed. [ ˆ ] b E b The varace of ˆb obaed by aga og ha from he lea quare deermg equao, ˆb ca be exreed a he followg lear combao of he obervao of Y. y x b ˆ x( x x ( é ê ë x ù ú y û By quarg ad ummg coa erm h lear combao, he varace foud o be 6

V [ bˆ ] ( x x x é x ê + ë ( x ù ú û Th exreo for he varace of he lea quare emae of he erce co of wo ar, he recrocal of he amle ze,, whch he uual facor for he varace of a mea, ad he rao of he quare of he mea of X o varace. A for b ˆ, a emae of he varace of ˆb ca be obaed by ubug he mea quared error emae of. ˆ ˆ b ( r SS é ù é Error x xy ( y x ê + ú ê + ë ( x û ë ( x Sce he redced value, ŷ, alo a lear combao of he lea quare arameer emae, oo wll be ormally drbued. ù ú û y ˆ é ê + ë ( x x( x x ( x ù ú y û The execed value of equao. ŷ obaed by drec ubuo o he lear redco [ yˆ ] E[ ˆ b ] + E[ ˆ b] x b + b x E A for ˆb ad ˆb, he varace of ŷ derved by quarg ad ummg he coa erm he exreo of ŷ a a lear combao of he obervao of Y. V [ yˆ ] æ ç ç + ç ç è ( x x ( x x ö ø ( x x æ ç + è ( x ö ø Aga oce ha he exreo for he varace of he redced value for he h obervao of Y co of wo comoe. The fr comoe, he recrocal of he amle ze, he uual facor for he varace of a mea. The ecod comoe a ormalzed quared dace of x, he h obervao of he exlaaory varable, from mea. Thu he varace of redco of Y for value of x ear mea wll be cloe o he varace of a ordary amle mea. Bu for value of x far from mea, he varace of he redco wll creae learly wh he quared ormalzed dace from he mea. 63

V [ yˆ ] Varace of Mea of Y 678 { + Normed D a ce From Mea of X 64748 ( x x ( x } The foregog exreo for he varace of ŷ he varace of he emae of he regreo le, whch he codoal mea of Y gve X. Bu he varace of a redco for gle obervao of Y a X wll be much greaer. Th redco error wll be he orgal varace of Y,, lu he varace of error due o emao of he regreo le. Therefore, he emaed varace of a gle obervao or redco a X ˆ ŷ ˆ { Sgle Obervao Re 64748 greo Le 4 } ( x x + + } ( x Boh emae of he regreo le ad redco from he regreo le wll be more accurae for value of X ear he mea..5. Iferece for he Regreo Le I ofe of ere o e hyohee abou he arameer of he regreo model or o coruc cofdece erval for varou quae aocaed wh he model. There may be heorecally recrbed value for he arameer. Cofdece erval for redco from he regreo model are frequely requred. Ifereal rocedure follow a aural way from he fac ha lea quare arameer emae ad hece he emaed regreo le all lear combao of he reoe varable, Y, ad le Y wll be ormally drbued. I addo, he emaed varace of hee arameer are derved from he um of quared devao of Y ad hece wll have a c drbuo. Therefore, he followg ac all have Sude' drbuo wh ( degree of freedom. bˆ b ˆ ˆ b bˆ b ˆ ( x Þ bˆ b ˆ ˆ b ˆ bˆ b é x ù ê + ( ú ë x û Þ ŷ b bx ˆ ŷ ˆ y b b x ( x x + ( x Þ where b ad b are he rue or hyohezed value of he regreo arameer. The mo commoly eed ull hyohe ha he loe ad erce equal zero. Th e he 64

e roduced by mo regreo ofware. For he loe arameer, h e ha a arcularly ereg erreao. ˆ b ˆ b ˆ r xy ( r ( r xy y x ( y x r xy xy ( whch mly he quare roo of he Fe for he regreo model derved earler. Thu, eg wheher he yemac comoe zero equvale o eg wheher he loe of he regreo le zero. If b, he he regreo le wll be horzoal a he mea of Y, ha, he mea of Y wll be redced a every value of X ad X wll have o effec o redco of Y. Cofdece erval for he erce, loe, regreo le, ad redco from he regreo le are calculaed he uual maer. ˆ b ± a, ˆ ( x bˆ ± ˆ é x ê + ë ( a, x ù ú û The cofdece erval for he regreo le bˆ + bˆ x ± ˆ a, ( x x + ( x Ad he cofdece erval for redco from he regreo le bˆ + bˆ x ± ˆ a, ( x x + + ( x The followg char dlay 95% cofdece erval for boh he regreo le ad dvdual redco from he regreo le. Noce ha he cofdece boud for he regreo le are very arrow ad clude very few of he orgal daa o. Th becaue he correlao bewee redcor ad crero varable hgh ad he f of he regreo le good. O he oher had, all orgal obervao are cluded wh he cofdece boud for gle o. 65

8 6 4 8 6 4 4 Cofdece Boud for Regreo Le Regreo Le Cofdece Boud for Sgle Po 4 6 8 Examle ad Soluo A comay rear deo calculaor ad cah reger. The daa from 8 rear are wre he able. Each rear ha mora e of daa. The fr oe a umber of reared calculaor (X ad he ecod oe a oal rear me (Y. 3 4 5 6 7 8 9 3 4 5 6 7 8 x 7 6 5 5 4 7 3 4 8 5 5 7 4 5 Y 97 86 78 75 6 39 53 33 8 65 5 7 5 7 49 68 a Fd arameer emae of he regreo le. b Draw daa ad regreo fuco. c Ue e for he value of all arameer of regreo fuco. Soluo you ca ue STATGRAPHIC ofware: 66

Lear regreo Rear me v. Number Regreo Aaly Lear model: Y b + b *x Deede varable: Rear Tme Ideede varable: Number Parameer Emae Sadard Error T Sac PValue b Ierce,35,56435,95549,3786 b Sloe 4,7383,5957 8,3834, b Ierce, b Sloe, he reul of hee value may be foud he ecod colum. The followg fuco roduce a equao for he emae of redced value: Rear Tme,35 + 4,7383. Number The oberved value of he e are how he fourh colum (T Sac ad correodg value are dlayed he la oe. I obvou ha hyohe H : β wll o be rejeced coderg he mora value value colum. Baed o h we ca ay ha regreo le ae hrough he begg wha a logcal cocluo, coderg he daa aure. The ecod of arcular e ay ha Sloe a value ha gfcaly dffer from zero ce we have rejeced H hyohe H : β. Regreo of he rear me Rear me 8 6 4 4 6 8 Number d Le fd he 95% cofdeal erval for he rear me deedece o he umber of calculaor. e Le fd o ad erval emao for a execed rear me for 5 calculaor. 67

Soluo Rear me 8 6 4 Regreo of he rear me 4 6 8 Number For value x5: Ŷ( æ ç ( x x b + b x ç + ç ç è ö x ( x x Y. ( x x ø 7 369 E(Yçx b + b x Î < Ŷ( x S (,Ŷ( x + S ( > < 69. 63, 73. 675 > Soluo SS SS + SS Y Ŷ a f Coder he qualy of examed model of lear regreo for he rear me deedece o a umber of calculaor ug a coeffce of deermao Ŷ R Ŷ a Source Sum of Square Regreo SS 68,6 Ŷ Error SS R 3,396 Toal SS 654, Y I SSŶ 98. 56 % SS Y 68

Summary Regreo model a ecal cae of geeral lear model. The bac aumo are deedece, homocedacy ad ormaly. Deede varable he varable of a regreo model ha radom ad we ry o decrbe ad exla behavor by mahemacal curve. Ideede (exlaaory varable are he varable he regreo model of whch behavor exla he behavor of he deede varable. Lear regreo model wh oe exlaaory varable a bac model ad baed o he LeaSquare Mehod. By h mehod model arameer ca be deermed. The um of quared devao of he real value from modelled value called he redual um of quare. We ca oba erval emao for he execed value of he deede varable. Thee erval boud form cofdeal erval of he regreo le. Quz. Decrbe ad exla equao of lear regreo.. Wha mea value he ANOVA able for lear regreo? 3. Wha roery decrbe a coeffce of deermao? 69

Praccal Exerce Exerce : Durg corol meaureme of dural comoe ze you radomly choe 8 comoe howg moly ove dvergece from ormal value he legh ad hegh: Legh dvergece [mm] 3 4 4 5 8 6 3 Hegh dvergece [mm] 4 6 5 6 7 3 9 4 Le fd he lear regreo model of deedecy bewee he legh dvergece ad hegh dvergece. {Awer: Ue a uable ofware acage.} Exerce : I he year 9396, waer flow rofle of Šace ad Moráva waer reervor were meaured. Average er year ( m 3 / are gve by he followg able: Year Šace Moráva Year Šace Moráva 93 4,3,476 946,68,374 93,386,35 947,45,94 933,576,38 948 3,543,799 934,466,75 949 4,55,4 935 3,576,8 95,4,9 936,8,93 95,74,55 937 3,863,354 95 3,79,99 938 3,76,68 953 3,87,488 939 3,7,534 954,677,83 94 4,49,38 955,86,878 94 4,466,57 956 3,8,4 94,584,76 957,59,65 943,38,63 958 3,656,87 944 3,7,8 959,447,38 945 3,9,43 96,77,679 Le aume ha oe of followg year, he average value of he whole year waer flow of Moráva reervor mg. I h year, he average waer flow for Šace reervor wa,9 m 3 /. Baed o lear regreo, ry o deerme he average waer flow Moráva reervor. {Awer: Ue a uable ofware acage.} 7

. SOLUTION KEYS.. Lecure. Exloraory daa aaly ofe a fr e revealg formao hdde large amou varable ad her vara.. The bac d of varable are quaave varable (omal, ordal ad qualave varable (dcree, couou. 3. The frequecy able cocered wh abolue ad relave frequece (for a qualave varable. 4. The ouler are he varable value whch gfcaly dffer from oher value. 5. b 6. qualave varable bar char, e char quaave varable box lo, em ad leaf lo 7. b 4 houad, d cca (9;9.. Lecure. P( A È B P( A + P( B P( A Ç B. P ( A Ç B P( A P( B 3. Two eve are deede f ereco robably of hee wo eve equal o a roduc of dvdual eve robable..3. Lecure 3+4. F( x P(X x < x x x ò. F( x f ( d for < x < 3. 5% quale called a meda A mode a value whch he dcree RV come wh he bgge robably. 4. The codoal drbuo he drbuo of oe varable a a fxed value of he oher joly drbued radom varable. 5. X X are muually deede Û F(x,,x F (x.. F (x. 6. The correlao coeffce meaure he regh of he relao bewee wo radom varable..4. Lecure 5. Dcree drbuo bomal, geomerc, egave bomal Couou drbuo oo, exoeal, Webull, Gamma. A equece of Beroull ral defed a a equece of radom eve whch are muually deede ad whch have oly wo oble oucome. The robably of 7

each oucome aumed o be he ame for all ral. O he ba of hee execao we ca defe he followg radom varable: bomal, geomerc ad egave bomal mea of he bomal radom varable: EX 3. A Gamma drbuo decrbe a me o h eve occurrece a Poo roce 4. β.5. Lecure 6. X... RV wh N(µ,σ m > Z X... N(,. Chebyhev' equaly u a uer boudary o he robably ha a obervao hould be far from mea. 3. The law of large umber a heorem abou covergece of mea he equece of he radom varable. 4. Chíquare drbuo a drbuo derved from he um of quared adard ormal radom varable..6. Lecure 7. Ifereal ac or acal duco comre he ue of ac o mae ferece cocerg ome uow aec of a oulao.. A radom amle a e of em ha have bee draw from a oulao uch a way ha each me a em wa eleced, every em he oulao had a equal ooruy o aear he amle..7. Lecure 8. The value calculao deed o defed ull hyohe: a H : m < m > valuef(x ob b H : m > m > valuef(x ob c H : m m > value{f(x ob, F(x ob }. I a hyohe ha acceed cae he rejeco of ull hyohe. ˆ ˆ X + X 3. Z... N(,, where ˆ æ ( ö + ˆ ˆ ç è ø.8. Lecure 9. A qˆ emao coe f a qˆ aymocally ubaed, E q q b lm Dq ˆ. P( T ( X q T ( X ³ a D H 7

.9. Lecure S B. F... Fdrbuo wh ( ad (N degree of freedom S W. ANOVA able 3. The o hoc aaly a ecod e of ANOVA ad co of comarg he mea of all ar of grou of uroe o chooe homogeou grou... Lecure. ya+b*x, where y a deede varable ad x a deede varable. The value a (erce ad b (loe are emae of regreo le arameer. x ad y are deede varable, he cae ha value >,5 3. Coeffce of deermao redcae abou uably of ued model 73