1. Fundamentals 1.1 Probability Theory Sample Space and Probability Random Variables Limit Theories

Transcription

1 1. undamentals 1.1 robablt Theor Sample Space and robablt Random Varables Lmt Theores

2 1.1 robablt Theor A statstcal model probablt model deals wth eperment whose outcome s not precsel repeatable even under supposedl dentcal condtons. -an eperment nvolvng chance Zuallseperment The ormulaton o a statstcal model requres two ngredents: - sample space Stchprobenraum - probablt Wahrschenlchket

3 1.1 robablt Theor perment o chance: A repeatable operaton under speced condtons whose outcome s not predctable - tossng a con - drawng card rom a complete set lementar outcomes rgebns enes perments: An elementar outcome s a possble outcome o an eperment o chance - n eperment tossng a con there are 2 elementar outcomes - n eperment drawng card there are 52 elementar outcomes

4 1.1 robablt Theor Sample space: The sample space o an eperment o chance Ω s the set o possble outcomes o the eperments - the sample space n the eperment tossng a con s head and tal vents regnsse: An event s a lst o the elementar outcomes. It can be consdered as a subset o the sample space o an eperment o chance. An event can be speced b { satses condton } - an event o the eperment drawng card : heart - the sample space s the largest event o an eperment o chance

5 1.1 robablt Theor Intersectons and Unons are operatons used to obtan new events Unon or Addton Denton: a pont s n the unon o event and event and onl t les ether n or n or possbl both: U + { or } {ω ω s n or } ropertes: + + commutatve [+]+G +[+G] assocatve Intersecton or product Denton: The ntersecton o two events s that event whose outcomes are those lng n both o the events: I { and }{ω ω s n both and } ropertes: []G [G] +G+G

6 1.1 robablt Theor More about operatons: Two events are sad to be dsjont or mutuall eclusve when ther ntersecton s empt: O Derence - o two events s gven b - {ω ω s n but not n } The complement o an event c s dened to be c Ω - One has + c Ω c O

7 1.1 robablt Theor robablt robablt o an event o a repeatable eperments s gven b Number o tmes when occurs N lm N Number o total trals N perment tossng a con : The relatve requenc o the event head up as the uncton o the number o trals

8 1.1 robablt Theor The probablt assgned to an event cannot be completel arbtrar and has to sats the ollowng aoms robablt aoms 1. Ω or ever event L or ever sequence o dsjont events 1 2L Consequences 1. c 1 2. or events

9 1.1 robablt Theor The dscrete case - the sample space has onl a nte or countabl nnte number o outcomes - the probablt or each ndvdual outcome s a nonnegatve number - the probablt o a gven event s gven b ω ω The contnuous case - the sample space s uncountabl nnte - the probablt o ndvdual outcomes s zero - t s necessar to assgn probabltes to events rather than to ndvdual ponts whereb denng the probablt o an event va probablt denst where Ω ω 1 ω dω ω dω

10 1.1 robablt Theor The addton law a general rule or determnng the probablt o a unon o two events n terms o the probabltes o these events The addton law s a consequence o Aom 3 whch s a specal addton or dsjont events. When decompose + nto three dsjont parts + + c + c one has accordng to Aom c + c [+ c ] + [+ c ] +

11 1.1 robablt Theor Condtonal robablt De bedngte Wahrschenlchket or an event contaned n an event o postve probablt the condtonal probablt o gven wrtten s dened to be when or n general Ω Ω new sample space ample: probablt or ran when temperature s larger than 20 C - cannot be zero - s proportonal to - satses the probablt aoms

12 1.1 robablt Theor The multplcaton law Independence vents and are sad to be ndependent and onl so that

13 1.1 robablt Theor Baes Theorem Usng the multplcatve rule to epress the probablt o an ntersecton as a product wth nether nor be zero elds Usng the Baes theorem s obtaned c c c + + c c +

14 1.1 robablt Theor robablt Space a probablt space Ω contans Ω as a sample space a collecton o subsets o Ω called events and the probablt assgned to the event Random Varable and Vectors A random varable vector s a measurable vector-valued uncton dened on a probablt space - as a uncton on a probablt space a random varable maps the events o Ω nto sets o values n the value space o the uncton and conversel - a random vector s a vector-values uncton on a probablt space whose components are random varables

15 1.1 robablt Theor ample I: An ndvdual tosses a con wth the outcomes beng sequences o heads and tals. The sample space contans all possble sequences. Some o the random varables are ωnumber o tosses requred to equalze the number o heads and the number o tals or the rst tme Yω1 or 0 dependng whether the thrd toss s heads or tals or the sequence HHTHTTTHT. ω6 Yω0 and or the sequence HTTTHTHHT ω8 Yω0 The probablt o ω 678 and the probablt o Yω0 can be worked out gven a ar con

16 1.1 robablt Theor ample II The pressure measured n Hamburg s a random varable. In ths case Ω R A possble value o the random varable s ω1003 ha Relatve to eample I t s now much harder to assess the probabltes o ω

17 1.1 robablt Theor The Dstrbuton uncton In general t s cumbersome to use the sample space and the probablt to descrbe the random characterstcs o a random varable. One use nstead the dstrbuton uncton. The dstrbuton uncton descrbes how the probablt s dstrbuted b a random varable n the space R o ts values. In partcular the cumulatve dstrbuton uncton c.d.. or dstrbuton uncton gves the amount assgned to the nterval rom up to λ [ ω λ] λ λ

18 1.1 robablt Theor Ω Gven a probablt space Ω and a random varable ω dened on t the correspondng dstrbuton uncton c.d. s completel determned as Ω ω Ths s uncton has the ollowng propertes s a non - decreasng uncton s contnuous rom the rght at each

19 1.1 robablt Theor A c.d. serves as a bass or computng the probablt o other event o nterest. or nstance the probablt o ab] can be computed n terms o the c.d.: ] ] ] ] - a b b a b a a b b a a b < + + or one has Snce

20 1.1 robablt Theor ractle or Quantle das Quantl o the c.d. The range o a dstrbuton s [01] and ths nterval can be dvded nto equal parts. I a c.d. rses steadl rom 0 to 1 wth no jumps or ntervals o constanc there s a unque number p or each p on the nterval [01] such that p p p s called a ractle quantle o the dstrbuton p Other notatons o p : the medan der Medan: p0.5 the kth quartle das k. Quantl: 4pk the kth decle das k. Dezl: 10pk the kth percentle das k. erzentl: 100pk

21 1.1 robablt Theor Notce: In the epresson λ s the dstrbuton uncton o reers to the whole uncton. Thus λ λ λ reer to the same uncton.. A derent dstrbuton uncton results rom usng a derent random varable λ λ λ Y λ Y Throughout ths course upper case e.g. reers a random varable whereas the correspondng lower case.e. ndcates a partcular value taken b the random varable

22 1.1 robablt Theor Dscrete Random Varables A random varable whose dstrbuton uncton jumps at values 1 2 and s constant between adjacent jump ponts s called dscrete. One has ample: p p p + L Three chps are drawn together at random rom a bowl contanng ve chps numbered and 5. There are 10 possble outcomes n ths eperment: ach havng probablt 1/10. Let ω denote the sum o the numbers on the chps n outcome ω. The values o are The correspondng probabltes or and 12 are

23 1.1 robablt Theor Contnuous Random Varables A random varable s sad to be contnuous. s a contnuous probablt measure on Ω. The c.d.. o a contnuous random varable s derentable almost everwhere. The dervatve o t s called the denst uncton 1 λ λ lmh 0 λ h h [ λ + h / 2 / 2 ]

24 1.1 robablt Theor The Denst uncton The probablt model or a contnuous dstrbuton can be dened b specng a denst uncton whch has the propertes d 1 The c.d.. can be constructed usng as ollows u du so does the probablt a < < b b a b a λ dλ

25 1.1 robablt Theor Analog between relatons n the dscrete and contnuous case 1 1 Contnuous Dscrete 1 d d d d d all + < < λ λ λ λ

26 1.1 robablt Theor Bvarate Dstrbutons A random vector ωyω ntroduces a probablt dstrbuton n the plane n the value space o the random vector. Ths dstrbuton s bvarate and gven b also reerred to as the jont dstrbuton uncton R 2 ] and [ and Y Y Ω ω ω The bvarate dstrbuton uncton satses to the as parallel or ever rectangle wth sdes respectvel and unctons o are unvarate dstrbuton and < + < k h k h k Y h

27 1.1 robablt Theor A bvarate dstrbuton s sad to be o dscrete tpe and Y A bvarate dstrbuton s sad to be o contnuous tpe the dstrbuton s contnuous and has a second-order med partal dervatve uncton rom whch can be recovered b The bvarate denst uncton satses 2 dvdu v u R dd

28 1.1 robablt Theor Margnal Dstrbutons Randvertelungen The ollowng dstrbutons are called the margnal dstrbutons o and Y and Y and Y Y In the dscrete case wth p j Y j the probablt that s obtaned b summng over j s or ed Y j j j p j In the contnuous case the margnal denst uncton s obtaned b derentatng the margnal dstrbuton uncton to obtan Y Y u ddu Y d

29 1.1 robablt Theor Condtonal Dstrbutons The new dstrbuton n the value space o a random varable gven event wth a postve probablt s called a condtonal dstrbuton and s dened b and A condtonal dstrbuton ma be contnuous n whch case ts denst s Or t ma be dscrete characterzed b d d and

30 1.1 robablt Theor The most commonl used condtonal dstrbutons have to do wth two varables and Y comprsng a bvarate random vector and the condton s then a condton on the value o the other varable e.g. I Y s contnuous the dstrbuton s dened through the condtonal denst Y Y Y Y Y du u and Y are ndependent random varables when Y Y Y Y

31 1.1 robablt Theor pectaton The epectaton o random varable µ s dened b or d or n general b d It holds + Y + Y a + b a + Y Y Y when s ndependent o Y

32 1.1 robablt Theor ample: Consder the eperment o three ndependent tosses o a ar con. Let ω denotes the number o heads n the sequence o e.g. HHT2. The sample ponts are HHH HHT HTH THH TTH THT HTT TTT and the value o are The probabltes o values o are 301/8 123/

33 1.1 robablt Theor Moments The k-th moment o a random varable s dened b and the k-th central moments b d k k k k or d k k k k or µ µ

34 1.1 robablt Theor Varance 2. Central Moment: the most requentl used hgh-order moment Var σ µ 2 Varance o sum Consder varance o the sum o random varables and Y Z+Y 2 Var Z Z Z Var + Var Y + 2cov Y Varance s not addtve! It holds n general 2 Var a + by a Var + b Var Y + 2abcov Y and n case and Y are ndependent Var Y Var + Var Y Var + Y Var + Var Y 2

35 1.1 robablt Theor Covarance and Correlaton 2. Moment between Two Random Varables The covarance between two random varables and Y s dened b cov Y µ Y µ Y Y Y and the correlaton b ρ Y cov Y σ σ Y One has or ndependent random varables and Y cov Y 0

36 1.1 robablt Theor Moments descrbe varous propertes o a dstrbuton - The mean rst moment: the center o gravt - The varance 2. Central moment: the spread -The skewness a scaled verson o the 3. Central moment: smmetr skewness γ 1 R γ 1 0: smmetrc d γ 1 <0 negatvel skewed or skewed to the let* γ 1 >0 postvel skewed or skewed to the rght 3 µ σ -The kurtoss a scaled and shted verson o 4. central moment: peakedness µ kurtoss γ 2 d 3 σ R 4 γ 2 <0: platkurtc less peaked than the normal dstrbuton γ 2 >0: leptokurtc more peaked than the normal dstrbuton * skewed let: the let tal s heaver than the rght tal

37 negatve skewed postve skewed

38 1.1 robablt Theor The Central Lmt Theorem I k k12 s an nnte seres o ndependent and dentcall dstrbuted random varables wth k µ and Var k σ 2 then the average 1 n k n k 1 s asmptotcall normall dstrbuted. That s lm n 1 n n k 1 k µ ~ 1 σ n N01

39 1.1 robablt Theor mprcal dstrbuton unctons o the amount o precptaton summed over a da a week a month or a ear at West Glacer Montana USA. The amounts have been normalzed b the respectve means and are plotted on a probablt scale so that a normal dstrbuton appears as a straght lne.