PROBABILITY AND STATISTICS FOR ENGINEERS

Transcription

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

2 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.

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

4 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

5 . 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

6 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

7 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

8 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 Clafcao 4 Clafcao Clafcao

9 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 Clafcao Clafcao 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

10 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.

11 We oberved car oal. Mode red (.e. our amle mo car were red Colour of ag car Colour of ag car 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

12 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.

13 Frequecy olygo for he evaluao grade frequecy 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 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

14 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

15 Grahcal ouu: Hogram Sold hr Pe Char Sold hr 8 7 abolue frequecy 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

16 .. 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

17 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 Soluo: a Mea: 7

18 I h cae we ue arhmecal mea: x x ! 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 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 ( ( ( ( ( ( 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: x ˆ

19 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 Daa acedg order Meda 4 Uer quarle 43 Lower quarle 5 The dfferece bewee he ad 3rd quarle called he IerQuarle Rage (IQR. 9

20 IQR x x.75.5 Examle: Daa Uer quarle 7 Lower quarle 4 IQR 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

21 .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

22 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 Now you ca dvde he daa e o quarle ad mar her varable value accordgly: Lower Quarle x,5 :.5; Þ z x 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 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 Þ x e. 75% muca are uder 43 (5% of hem are 43 year old or older. b IerQuarle Rage IQR:

23 IQR x.75 x 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 MAD M.5.5; Þ z x 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

24 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 / / / 3/ 9 35 / 5/ 47 4 / 6/ 4 4 / 7/ / 8/ / 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,, x he mo commo meaure of varably 4

25 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

26 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 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

27 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

28 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 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 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

29 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 α α> α< Kuro Kuro reree cocerao of varable value aroud her mea. The followg formula ued o ge value: b ( + ( ( ( 3 ( x x ( ( ( 3 Kuro erreao: 9

30 b... Kuro correod o ormal drbuo b >... "eaed" drbuo of he varable b <... "fla" drbuo of he varable 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 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

31 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 * 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 * ( Sem wdh Leave Cumulave frequece 3

32 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 * 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

33 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 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: Draw box lo, emrcal drbuo fuco ad calculae he mea, adard devao, horh, mode ad erquarle rage. 33

34 . 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

35 .. 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

36 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

37 .. 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

38 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

39 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

40 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

41 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

42 .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

43 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

44 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

45 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} 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

46 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 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 47

48 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

49 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 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,

50 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 , 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 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 ] They aed he mah They faled he mah Toal exam exam They aed he hyc exam 8 They faled he hyc exam Toal 38 ad you calculae ad fll he remag daa: 5

51 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: Ad for he umber of ude who aed boh exam: They aed mah exam They faled o a mah exam They aed hyc exam They faled o a hyc exam Toal 8 38 The robable are: P ( M Ç F ; P( F, whch mle ha: ( M Ç F P P( M F P( F 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

52 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 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

53 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

54 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

55 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

56 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 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

57 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 / x 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

58 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

59 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 f(x

60 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

61 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 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

62 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

63 a 3 >... ovely ewed e 6. Kuro a 4 m 4 / x 4 I a level of uro (flae: a 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