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BAKAL´AˇRSK´A PR´ACE Statistika a Ztroskotán´I Titaniku

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BAKAL´AˇRSK´A PR´ACE Statistika a Ztroskotán´I Titaniku UNIVERZITA PALACKEHO´ V OLOMOUCI PRˇ´IRODOVEDECKˇ A´ FAKULTA KATEDRA MATEMATICKE´ ANALYZY´ A APLIKAC´I MATEMATIKY BAKALA´ RSKˇ A´ PRACE´ Statistika a ztroskot´an´ıTitaniku Vedouc´ıbakal´aˇrsk´epr´ace: Vypracoval: RNDr. Miloslav Z´avodn´y Vojtˇech Sukaˇc Rok odevzd´an´ı:2012 MATEKO, III. roˇcn´ık Prohl´aˇsen´ı Prohlaˇsuji, ˇze jsem vytvoˇril tuto bakal´aˇrskou pr´aci samostatnˇeza veden´ıRNDr. Miloslava Z´avodn´eho a ˇze jsem v seznamu pouˇzit´eliteratury uvedl vˇsechny zdroje pouˇzit´epˇri zpracov´an´ıpr´ace. V Olomouci dne 15. dubna 2012 Podˇekov´an´ı R´ad bych na tomto m´ıstˇepodˇekoval vedouc´ımu bakal´aˇrsk´e pr´ace RNDr. Miloslavu Z´avodn´emu za obˇetavou spolupr´aci i za ˇcas, kter´ymi vˇenoval pˇri konzultac´ıch. D´ale si zaslouˇz´ıpodˇekov´an´ım˚uj poˇc´ıtaˇc, ˇze vydrˇzel moje pracovn´ıtempo, a typografick´ysyst´em LATEX, kter´ym je pr´ace vys´azena. Velk´yd´ıkpatˇr´ı tak´evˇsem, kteˇr´ıse mnou ochotnˇekonzultovali pˇri psan´ı t´eto pr´ace a vˇsem, kteˇr´ımi nahl´asili pˇreklepy a chyby. Obsah Obsah 3 Uvod´ 5 1 Teorie 6 1.1 Teoriepravdˇepodobnosti. .......... 6 1.2 Matematick´astatistika. ........ 12 1.3 Kontingenˇcn´ıtabulky . .. .. .. .. .. .. .. .. .. .. .. .. ........ 13 1.4 Popisn´astatistika.................................. ....... 16 1.4.1 Jednoduch´a(p´arov´a)korelace . .......... 16 1.4.2 V´ıcen´asobn´akorelace. ....... 20 1.4.3 Parci´aln´ı(d´ılˇc´ı)korelaˇcn´ıpomˇer . ............. 20 2 Cetnostiˇ 23 2.1 Element´arn´ıˇcetnosti . ........... 24 2.2 Cetnostisjedn´ımpromˇenn´ymindexemˇ . ......... 24 2.3 Cetnostisv´ıcepromˇenn´ymiindexyˇ . .......... 26 2.4 Cetnostiobsahuj´ıc´ıkapit´ana.ˇ . ........... 29 3 V´ypoˇcty pravdˇepodobnost´ıpˇreˇzit´ı 30 3.1 V´ypoˇcty s element´arn´ımi podm´ınkami . ............. 30 3.2 V´ypoˇcet pˇri sdruˇzen´ych ˇcetnostech pˇres jeden index ..................... 33 3.3 V´ypoˇcet pˇri sdruˇzen´ych ˇcetnostech pˇres v´ıce index˚u...................... 40 3.4 Shrnut´ı ......................................... ..... 45 4 Kontingenˇcn´ıtabulky 46 4.1 Zkoum´an´ız´avislosti na m´ıstu nalodˇen´ı . ............... 46 4.2 Zkoum´an´ız´avislostinapohlav´ı . ............ 47 4.3 Zkoum´an´ız´avislostina um´ıstˇen´ı . .............. 47 5 Korelaˇcn´ıpoˇcet 49 5.1 P´arov´akorelace.................................. ........ 49 5.2 V´ıcen´asobn´akorelace. .......... 51 6 Nov´akategorizace z´akladn´ıho souboru 52 6.1 Cetnosti...........................................ˇ ... 52 6.2 Pravdˇepodobnosti ................................ ........ 53 6.3 Kontingenˇcn´ıtabulky . .. .. .. .. .. .. .. .. .. .. .. .. ........ 56 6.3.1 Z´avislost na m´ıstu nalodˇen´ı . ....... 56 6.3.2 Z´avislost na pohlav´ı . .... 56 6.3.3 Z´avislost na um´ıstˇen´ı. ....... 56 6.4 Korelaˇcn´ıpoˇcet................................. ......... 57 6.4.1 Jednoduch´a(p´arov´a)korelace . .......... 57 6.4.2 V´ıcen´asobn´akorelace. ....... 58 3 Z´avˇer 61 Literatura 63 Pˇr´ılohy 64 TitanicPassengersandCrew . ........ 65 Victims of the Titanic Disaster . ....... 90 SurvivorsoftheTitanicDisaster . ......... 107 4 Uvod´ C´ılem t´eto pr´ace je zpracovat velk´ystatistick´ysoubor. Jelikoˇztento soubor je tvoˇren pˇr´ım´ymi ´uˇcastn´ıky nejvˇetˇs´ın´amoˇrn´ıkatastrofy v dˇejin´ach, nab´ız´ıse jako c´ıl: • urˇcit pravdˇepodobnost pˇreˇzit´ıurˇcit´eskupiny osob se stejn´ymi nebo podobn´ymi charakteristikami, • zkoumat z´avislost pravdˇepodobnosti pˇreˇzit´ına tˇechto charakteristik´ach, • prok´aˇze-li se z´avislost, vhodnˇeji popsat. Pˇri zpracov´an´ısouboru budeme pouˇz´ıvat pouze: • poˇc´ıtaˇcs tabulkov´ym kalkul´atorem, • tuˇzku a pap´ır. Tedy ˇz´adn´ysofistikovan´ystatistick´ysoftware. Na tomto m´ıstˇejeˇstˇezm´ın´ıme p´ar z´akladn´ıch a obecnˇezn´am´ych informac´ıo Titaniku. Poˇc´atkem dubna roku 1912 (tedy pˇresnˇepˇred sto lety) vyplul Titanik z pˇr´ıstavu v Belfastu. Na palubˇebylo skoro 200 ˇclen˚upos´adky a deset pasaˇz´er˚u, t´emˇeˇrvˇsichni cestuj´ıc´ıbyli pracovn´ıci lodˇenice. V Southamptonu, dne 10. dubna na palubu Titaniku nastoupili dalˇs´ıpasaˇz´eˇri a zbytek pos´adky vˇcetnˇe kapit´ana Edwarda Johna Smitha. Pot´eTitanik opustil Southamptonsk´ypˇr´ıstav a zam´ıˇril k francouzsk´emu Cherbourg a n´aslednˇek irsk´emu Queenstownu. Dne 11. dubna roku 1912 Titanik opustil Queenstown s 2 208 lidmi na palubˇea poprv´ea naposledy vyplul na oce´an, kde o p´ar dn´ıpozdˇeji, v nedˇeli 14. dubna 1912 asi 20 minut pˇred p˚ulnoc´ınarazil do ledov´ehory, coˇzse mu stalo osudn´ym a 15. dubna 1912 ve 2:20 hodin klesla ´udajnˇenepotopiteln´alod’ pod hladinu oce´anu. Celkem se na Titaniku plavilo 2 240 lid´ı,nˇekteˇr´ız nich vˇsak vystoupili dˇr´ıve, neˇzTitanik vyplul od pˇr´ıstavu v Queenstownu. Pozn´amka. Vzhledem ke skuteˇcnosti, ˇze r˚uzn´eprameny se liˇs´ıv poˇctu obˇet´ıa pˇreˇzivˇs´ıch, a to nejen v jejich pomˇeru, ale tak´ev souˇctu, vych´az´ıme z jedin´eho zdroje a to z tabulky [1]. Rozhodnut´ıo tom, zda dotyˇcn´aosoba pˇreˇzila ˇci nikoliv, provedeme na z´akladˇe porovn´an´ıt´eto tabulky s tabulkami [2] a [3]. 5 Kapitola 1 Teorie Jelikoˇzv tomto textu budeme pouˇz´ıvat pojmy a poznatky z teorie pravdˇepodobnosti, matematick´e statistiky a popisn´estatistiky, je potˇreba je nyn´ınadefinovat. 1.1 Teorie pravdˇepodobnosti Pokusem rozum´ıme uskuteˇcnˇen´ıurˇcit´eho pevnˇedan´eho syst´emu podm´ınek. V teorii pravdˇepodob- nosti se studuj´ıv´ysledky tzv. n´ahodn´ych pokus˚u. Tyto pokusy konˇc´ınastoupen´ımpr´avˇejednoho v´ysledku z mnoˇziny moˇzn´ych v´ysledk˚uΩ. Ω je tedy mnoˇzina vˇsech moˇzn´ych v´ysledk˚un´ahodn´eho pokusu, ω ∈ Ω je jeden konkr´etn´ıv´ysledek. Mnoˇzina Ω m˚uˇze b´yt jednoprvkov´a(pak hovoˇr´ıme o deterministick´em pokusu), alespoˇndvouprvkov´a, koneˇcn´anebo nekoneˇcn´a.1 Pozn´amka. V naˇsem pˇr´ıpadˇebude ωj znamenat pokus je proveden na j-t´eosobˇe. Definice 1 2 Kaˇzd´amnoˇzina A ⊂ Ω se naz´yv´a jev, jednoprvkov´epodmnoˇziny naz´yv´ame element´arn´ı jevy. ⋄ Jevy budeme oznaˇcovat velk´ymi p´ısmeny ze zaˇc´atku abecedy: A, B, C,... Symbol ⊂ budeme uˇz´ıvat pro vlastn´ıi nevlastn´ıinkluzi. Definice 2 3 R´ık´ame,ˇ ˇze pˇri realizaci n´ahodn´eho pokusu nastal jev A, nastal-li v pokusu takov´yv´ysledek ω, pro kter´yplat´ı ω ∈ A. Je to tzv. v´ysledek pˇr´ızniv´yjevu A. Pr´azdn´amnoˇzina ∅ se naz´yv´a nemoˇzn´yjev (nenastane pˇri ˇz´adn´erealizaci n´ahodn´eho pokusu), mnoˇzina vˇsech v´ysledk˚un´ahodn´eho pokusu Ω se naz´yv´a jist´yjev (nastane pˇri kaˇzd´erealizaci pokusu). ⋄ 1ispirov´ano [Kunderov´a(2004), s. 7] 2[Kunderov´a(2004), s. 8] 3[Kunderov´a(2004), s. 8] 6 4 Definice 3 Sjednocen´ıjev˚u A1,..., An je jev, kter´ynastane pr´avˇetehdy, nastane-li aspoˇnjeden z jev˚u A1,..., An. Znaˇc´ıse n A1 ∪ ... ∪ An nebo Aj . j=1 [ A A ∞ A Sjednocen´ım˚uˇze obsahovat i nekoneˇcn´ypoˇcet jev˚u. Je-li jich spoˇcetnˇemnoho 1, 2,..., p´ıˇseme n=1 n. S Pr˚unik jev˚u A1,..., An je jev, kter´ynastane pr´avˇetehdy, nastanou-li vˇsechny tyto jevy (spoleˇcnˇe, souˇcasnˇe). Znaˇc´ıse n A1 ∩ ... ∩ An nebo Aj . j=1 \ Pr˚unik m˚uˇze obsahovat i nekoneˇcn´ypoˇcet jev˚u. Tvoˇr´ı-li tyto jevy nekoneˇcnou posloupnost, p´ıˇseme ∞ A n=1 n. T Jev opaˇcn´yk jevu A (doplnˇek, komplement jevu A) je jev, kter´ynastane pr´avˇetehdy, kdyˇznenastane A. Budeme jej oznaˇcovat Ac, tedy Ac =Ω \ A. Rozd´ıl jev˚u A, B je jev, kter´ynastane pr´avˇetehdy, nastane-li A a z´aroveˇnnenastane B. Znaˇc´ıme jej A \ B, tedy A \ B = A ∩ Bc. Rekneme,ˇ ˇze jev B je d˚usledkem jevu A (A m´aza n´asledek B, A implikuje B), je-li kaˇzd´yv´ysledek, kter´yje pˇr´ızniv´yjevu A, tak´epˇr´ızniv´yjevu B, tedy plat´ı-li A ⊂ B. (R´ık´aseˇ tak´e, ˇze jev A je podjevem jevu B). Jevy A, B nazveme nesluˇciteln´e (disjunktn´ı), nemohou-li nastat souˇcasnˇe, tj. je-li A ∩ B = ∅. Rekneme,ˇ ˇze jevy A1,..., An, resp. A1, A2,..., Aj 6= ∅, ∀j = 1, 2,...,n, resp. ∀j = 1, 2,..., tvoˇr´ı rozklad jevu B, jsou-li tyto jevy disjunktn´ıa plat´ı-li n ∞ A = B resp. A = B j n ⋄ j=1 n=1 [ [ Pro jevy tak´eplat´ıvˇsechna tvrzen´ı,kter´aplat´ıpro mnoˇziny, zejm´ena pak A c Ac A c Ac 1. de Morganova pravidla: i i = i i , i i = i i , 2. A ∩ (B \ C) = (A ∩ B) \ (AS∩ C), T T S A B A B 3. j j ∩ = j ( j ∩ ). S S Pozn´amka. V naˇsem pˇr´ıpadˇejev A = {ωA}⊂ Ω, kde {ωA} je mnoˇzina takov´ych v´ysledk˚u ωj , kde j-t´a osoba pˇreˇzila. 4inspirov´ano [Kunderov´a(2004), s. 8] 7 Definice 4 5 Necht’ Ω 6= ∅ je libovoln´amnoˇzina. Nepr´azdn´ysyst´em A podmnoˇzin mnoˇziny Ω se naz´yv´a jevov´epole, plat´ı-li a) A ∈ A ⇒ Ac ∈ A, A ∞ A b) n ∈ A, n =1, 2,... ⇒ 1 n ∈ A. Prvky A ∈ A se naz´yvaj´ı n´ahodn´ejevy S . ⋄ Pozn´amka. V dalˇs´ımtextu budeme v´yrazem jev rozumˇet n´ahodn´yjev. Definice 5 6 Necht’ je d´ana nepr´azdn´amnoˇzina Ω a na n´ıjevov´epole A. Pravdˇepodobnost´ı nazveme kaˇzdou re´alnou funkci P( ) definovanou na A, kter´avyhovuje n´asleduj´ıc´ımaxiom˚um: a1: P(Ω) = 1, a2: P(A) ≥ 0, ∀A ∈ A. a3: Pro libovolnou posloupnost An ∈ A, n =1, 2,... nesluˇciteln´ych n´ahodn´ych jev˚uplat´ı ∞ ∞ P A P A n = ( n). ⋄ 1 1 [ X Definice 6 7 Uspoˇr´adanou trojici (Ω, A, P) naz´yv´ame pravdˇepodobnostn´ıprostor. ⋄ 8 Definice 7 Mˇejme nepr´azdnou mnoˇzinu Ω = {ω1,...,ωn} a na n´ıjevov´epole A, kter´eobsahuje vˇsechny podmnoˇziny mnoˇziny Ω. Re´aln´afunkce P( ) definovan´ana A pˇredpisem 1 P {ω } = , j =1,...,n j n P j:ωj ∈A {ωj} n nA P(A)= P {ωj} = · = , ∀A ∈ A, P {ωj} n n j:ω ∈A Pj:ωj ∈Ω Xj P kde nA je poˇcet tˇech v´ysledk˚un´ahodn´eho pokusu, kter´ejsou pˇr´ızniv´ejevu A,a n je poˇcet vˇsech v´ysledk˚u n´ahodn´eho pokusu, se naz´yv´a ˇcetnostn´ıpravdˇepodobnost.
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