Probability and Statistics. What is probability? What is statistics?

Transcription

1 robablt ad Statstcs What s robablt? What s statstcs?

2 robablt ad Statstcs robablt Formall defed usg a set of aoms Seeks to determe the lkelhood that a gve evet or observato or measuremet wll or has haeed What s the robablt of throwg a 7 usg two dce? Statstcs Used to aalze the frequec of ast evets Uses a gve samle of data to assess a robablstc model s valdt or determe values of ts arameters After observg several throws of two dce ca I determe whether or ot the are loaded Also deeds o what we mea b robablt

3 robablt ad Statstcs We erform a eermet to collect a umber of to quarks How do we etract the best value for ts mass? What s the ucertat of our best value? Is our eermet terall cosstet? Is ths value cosstet wth a gve theor whch tself ma cota ucertates? Is ths value cosstet wth other measuremets of the to quark mass? 3

4 robablt ad Statstcs 4

5 robablt ad Statstcs etaquark search - how ca ths occur? σ effect 005 o effect 5

6 robablt Let the samle sace S be the sace of all ossble outcomes of a eermet Let be a ossble outcome The ( foud [d]) f()d f() s called the robablt dest fucto (df) It ma be called f(;θ) sce the df could deed o oe or more arameters θ Ofte we wll wat to determe θ from a set of measuremets Of course must be somewhere so f ( ) d 6

7 robablt Deftos of mea ad varace are gve terms of eectato values E [] f ( ) d μ V [ ] [ ] E μ σ [] E ( E[] ) 7

8 robablt Deftos of covarace ad correlato coeffcet V ρ f cov ad the cov σ σ ad so cov [ ] E[ ( μ )( μ )] E[ ] [ ] deedet the E [ ] f ( ) [ ] 0 f ( ) f ( ) f ( ) dd μ μ μ μ 8

9 9 robablt Error roagato ( ) ( ) [ ] [ ] ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] j j j j j V E E V E we fd the TS eadg cov ad ad... wth varables Cosder μ μ σ μ μ μ μ r r r r r r r r r r r r r

10 0 robablt Ths gves the famlar error roagato formulas for sums (or dffereces) ad roducts (or rato) [ ] [ ] [ ] ( ) [ ] [ ] cov ad for cov we fd for Usg E E V σ σ σ σ σ σ

11 Uform Dstrbuto Let What s the osto resoluto of a slco or multwre roortoal chamber wth detecto elemets of sace? [] ( ) ( ) [] [] ( ) [ ] [] ( ) ( ) otherwse for 0 ) ; ( α β α β β α β α α β α β β α β α β α d V E E V d d f E β α f

12 Bomal Dstrbuto Cosder deedet eermets (Beroull trals) Let the outcome of each be ass or fal Let the robablt of ass robablt of But there are f ( ; )!! successes! ( )! ( )!! ( ) ( ) ermuatos for dstgushable objects groug them at a tme

13 ermutatos Quck revew umber of ermutatos for elemets! Ths cosders each elemet dstgushable But elemets of the frst te are dstgushable so!of the elemets lead to the same stuato Dtto for the remag ( -) Thus accoutg for these rrelevat ermutatos leads to umber of uque ermuatos s elemets of! ( -)!! the secod te 3

14 4 Bomal Dstrbuto For the mea ad varace we obta (usg small trcks) Ad ote wth the bomal theorem that [] ( ) [] [ ] [] ( ) ( ) E E V f E ; 0 ( ) ( ) f 0 0 ) ; (

15 Bomal Dstrbuto Bomal df 5

16 Eamles Bomal Dstrbuto Co fl (/) Dce throw (/6) Brachg rato of uclear ad artcle decas (Br) Detector or trgger effceces (ass or ot ass) Blood grou B or ot blood grou B 6

17 Bomal Dstrbuto It s baseball seaso! What s the robablt of a htter gettg 4 hts oe game? f E V ( 4;40.3) [] [] ! 0.3 4!0! Eect. ± 0.84 hts for a htter 7

18 osso Dstrbuto Cosder whe 0 E E V [] v f ( ; v) e! ad oe fds [] v The osso df v [] σ v v s 8

19 9 osso Dstrbuto ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) e v e f e e f v v v ad small for large!! ;...!...! for large!!!!! ;

20 osso df osso Dstrbuto 0

21 Eamles osso Dstrbuto artcles detected from radoactve decas Sum of two osso rocesses s a osso rocess artcles detected from scatterg of a beam o target wth cross secto σ Cosmc ras observed a tme terval t umber of etres a hstogram b whe data s accumulated over a fed tme terval umber of russa solders kcked to death b horses Ifat mortalt QC/falure rate redctos

22 osso Dstrbuto Let 0.7! (;) 0.7! (;) ! (0;) ad / The 0 0 Let atoms 0 Let e e e s v /s ~ τ ~

23 Gaussa Dstrbuto Gaussa dstrbuto Imortat because of the cetral lmt theorem For deedet varables that are dstrbuted accordg to a df the the sum wll have a df that aroaches a Gaussa for large Eamles are almost a measuremet error (eerg resoluto osto resoluto ) E V [ ] [ ] μ σ 3

24 Gaussa Dstrbuto The famlar Gaussa df s f E V ( ; μ σ ) [] μ [] σ πσ e ( μ) σ 4

25 Gaussa Dstrbuto Some useful roertes of the Gaussa dstrbuto are ( rage μ±σ) ( rage μ±σ) ( rage μ±3σ) ( outsde rage μ±3σ) ( outsde rage μ±5σ) ( rage μ±0.6745σ) 0.5 5

26 χ Dstrbuto Ch-square dstrbuto / f ( z; ) z / Γ /... s [] [] z E z V The usefuless z deedet ( μ ) follows the σ ( ) the umber of χ of ths df μ σ e z degrees s / dstrbuto wth ( z 0) of that for d.o.f. freedom 6

27 χ Dstrbuto 7

28 robablt 8

29 robablt robablt ca be defed terms of Kolmogorov aoms The robablt s a real-valued fucto defed o subsets AB samle sace S For ever subset If A B ( S) 0 A S ( A) ( A B) ( A) ( B) Ths meas the robablt s a measure whch the measure of the etre samle sace s 0 9

30 robablt We further defe the codtoal robablt (A B) read (A) gve B ( A B) Baes theorem Usg ( A B) ( A B) ( B) ( A B) B( B A) ( B A) ( A) ( B) 30

31 robablt For dsjot A ( ) ( ) ( ) B the ( ) A B B A ( B A) ( A) ( B A ) ( A ) Usuall oe treats the A as outcomes of a reeatable eermet A 3

32 robablt Usuall oe treats the A as outcomes of a reeatable eermet The (A) s usuall assged a value equal to the lmtg frequec of occurrece of A A ( A) lm Called frequetst statstcs But A could also be terreted as hotheses each of whch s true or false The (A) reresets the degree of belef that hothess A s true Called Baesa statstcs 3

33 Baes Theorem Suose the geeral oulato (dsease) 0.00 (o dsease) Suose there s a test to check for the dsease ( dsease) 0.98 (- dsease) 0.0 But also ( o dsease) 0.03 (- o dsease) 0.97 You are tested for the dsease ad t comes back. Should ou be worred? 33

34 Baes Theorem Al Baes theorem ( dsease ) ( dsease ) ( dsease) ( dsease) ( dsease) ( dsease) ( o dsease) ( o dsease) % of eole testg ostve have the dsease Your degree of belef about havg the dsease s 3.% 34

35 Baes Theorem Is athlete A gult of drug dog? Assume a oulato of athletes ths sort (drug) (o drug) Suose there s a test to check for the drug ( drug) 0.99 (- drug) 0.0 But also ( o drug) (- o drug) The athlete s tested ostve. Is he/she volved drug dog? 35

36 36 Baes Theorem Al Baes theorem??? ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ad ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( o drug drug drug o drug o drug o drug o drug o drug drug o drug o drug drug drug drug drug drug