B a s i c. C o u n t

Transcription

1 B a s c C o u n t n g S t a t s t c s

2 St ochast c uclear Observables St at st cs sour ces of st ochast c obser vables n nuclear scence: ) uclear phenoena ar e gover ned by quant al wave f unct ons and nherent statstcs ) Det ect on of pr ocesses occur s wt h per f ect ef f cency ( < ) and f nt e r esolut on dst r but ng shar p event s over a range n. Stochastc observables have a r ange of values wt h f r equences det er ned by pr obablt y dst r but on P(). char act er ze by set of oent s of P < n > = n P ()d; n,,, wth the noralzaton < > =. Frst oent of P: E() = <> = P ()d second cent r al oent = var ance of P(): = < - < > >

3 Uncert ant y and St at st cs ucleus s a quant al syst e descr bed by a wave f unct on (, ;t ) (, ;t ) ar e t he degr ees of f r eedo of t he syst e and t e. Probablty denst y (e.g., f or, nt egr at e over ot her s) dp(, t), t d oralzaton dp(, t) P t d d t d (, ), Transton between states M E dp (, t) d t t, ( ) e e state dsappears Probablty rate f or dsappear ance (decay of ) can var y over any or der s of agnt ude no certanty

4 The oral Dst rbut on..5 Contnuous functon or dscrete dstrbuton (over bns) St at st cs 4 G ( ) P() G ( ) G n P() <> oral (Gaussan) Probablty P( ) ep FWHM X ln.35 or alzed pr obablt y P( ) d ep X X X n 7

5 Eperent al Mean Count s and Varance St at st cs 5 What can be easur ed: enseble (saplng) aver ages (epect at on values) and uncertantes 36 U (.5g) sour ce, count par t cles et t ed dur ng = t e nt er vals n). =?? Average count n n a saple : n n-<n> (n-<n>) E <n> <n-<n>> n n n ( n unknown) populaton Varance of n n the ndvdual saples s n n ( ) Varance (" error ") of the saple average n s populaton n ( n n ) ( ) n Std devaton n. : n n / 5 Result : n n ( ) n pop 9 sae for saple tes as large

6 P() 3 Saple St at st cs Assue t r ue populat on dst r but on f or var able P( ) ep v X v X pop St at st cs 6 5 orally Dstrbuted Events 5.5 wt h t r ue ( populat on ) ean <> pop = 5., =. 5 Result s of 3 saples (easur eent s): orally Dstrbuted Events Saple aver age <> = ( )/ 3 = 5. 5 orally Dstrbuted Events <> =5. =. <> =4.96 =.3 <> =4.96 = Saple varance s = = [(5.-5. ) +( ) ]/ =. s =.75 =.75/3 =.5 =.5 Result: <> pop 5. ±.5 <>+s <> <>-s

7 Eaple Saple sze St at st cs 7 n = n = 5 FRAME orally Dstrbuted Events 5 5 MC Events n bns orally Dstrbuted Events MC Events n bns The larger the saple, the narrower the dstrbuton of values, the ore t approaches the true Gaussan (noral) dstrbuton. 5 5

8 Cent ral- Lt Theore The eans (aver ages) of df f er ent saples n t he pr evous eaple clust er t oget her closely. general property: St at st cs 8 The dst r but on of t he saple eans appr oaches a Gaussan nor al dst r but on, f t he sze n of t he saple ncr eases, r egar dless of t he f or of t he or gnal (populat on) dst r but on. The aver age of a dst r but on does not cont an nf or at on on t he shape of t he dst r but on. The aver age of any t r uly r ando saple of a populat on s alr eady close t o t he t r ue populat on aver age. Consder ng any saples, or lar ge saples, nar r ows t he choces. The Gaussan wdt h becoes nar r ow. The st andar d er r or of t he ean decr eases as t he saple sze ncreases.

9 Bnoal Dst rbut on I nt eger r ando var able = nuber of events, out of total, of a gven type, e.g., decay of (fro a saple of ) radoactve nucle, or detecton of (out of ) phot ons ar r vng at det ect or. p = pr obablt y f or a (one) success (decay of one nucleus, det ect on of one photon) St at st cs 9 Choose an ar bt r ar y saple of trals out of trals p = probablty for at least successes (- p) - = probablty for - f alur es (sur vvals, escapng det ect on) Probablty for eactly successes out of a total of trals P ( ) p p How any ways can event s be chosen out of? coeffcent! ( )!( )! Bnoal Tot al pr obablt y (success r at e) f or any saple of events: P p p bnoal ( )

10 St at st cs Pbnoal( ) p p.36 Pb(.) Pb(.3) Moent s and Lt s Bnoal Dstrbutons =3 Pr obablt y f or successes out of trals, ndvdual probablty p oralzaton P ( ) p p bnoal Mean and varance p and p( p) Dst r but ons f or =3 and p=. p=.3 Posson Gaussan p( p) p l Pbnoal( ) ep

11 Posson Probablt y Dst rbut on St at st cs Results fro bnoal dst r but on n t he lt of sall p and large ( p > ) l P (, ) P (, ) p and.67 Pp (.5 ) Pp ( 3 ) Pp ( 5 ) Pp ( ) bnoal Posson Dstrbutons 5 5 Posson Pr obablt y f or obser vng events when average s <> = P Posson (, ) e! =,,, = <> = p and = s the ean, the average nuber of successes n trals. Observe counts (events) uncertanty s = Unlke t he bnoal dst r but on, the Posson dstrbuton does not depend eplctly on p or! For large, p: Posson Gaussan (oral Dstrbuton)

12 Moent s of Transt on Probablt es St at st cs 3 6. :.5g.5g g 4 n n p n Probablty for decay ( decay rate per nucleus) : p n 4 7 corresponds to " half lfe" t/.34 a 7 4 Sall pr obablt y f or pr ocess, but any t r als (n = ) < n < St at st cal process f ollows a Posson dst rbut on: n= rando Df f erent st at st cal dst rbut ons: Bnoal, Posson, Gaussan

13 Radoact ve Decay as Posson Process St at st cs 3 Usef ul when only a ean count r at e s known: decay, backgr ound count s, or r eact on. 37 Cs unst able sot ope, decays t / = 7 year s p = ln/ 7 =.6/ a = 8. - s - Saple of g: = 5 nucle (=t r als f or decay) How any wll decay? = p = s - Count r at e est at e d/ dt = (8. +5 ± 95) s - Pr obablt y f or decays P (,) = est at ed P Posson (, ) e (8.5 )!! e

14 Funct ons of St ochast c Varables Rando ndependent var ables,,., n cor r espondng var ances,,., n Funct on f (,,., n ) St at st cs 4 Gauss law of er r or pr opagat on: f... n n f f f f f.. f, 3,.., 3,..,,.., n Further ters f not ndependent ( correlatons) Ot her wse, ndvdual coponent var ances ( f) add.

15 Eaple: Spect ral Analyss Addng or subt r act ng Posson dst r but ed nuber s and : Varances always add : St at st cs 5 B B Peak Area A Background B Analyze peak n range channels c c : begnnng of backgr ound lef t and rght of peak n = c c +. Total area = (c )=B, (c )=B, Lnear background B = n(b +B )/ Peak Area A: A n B B / A n B B / 4

16 St at st cs 6 Measur ed Pr obablt y 3 Conf dence Level Assue nor ally dst r but ed observable : P( ) ep v P( pop ) ep d CL Wt h conf dence level CL (pr obablt y n %), t he t r ue value < pop > dffers by less than = n fro easured average. Trustworthy eptl. CL( ) 68.3% CL( ) 95.4% results quote ±3 CL( 3 ) 99.7% error bars! pop v pop Saple dst r but on wt h dat a set observed average <> and std. error approate populaton. Confdence level CL (Cent r al Conf dence I nt er val): pop

17 Set t ng Conf dence Lt s Eaple: Sear ch f or r ar e decay wt h decay r at e obser ve no count s wt hn t e t. Decay pr obablt y law dp/ dt ep {- t }. Law s syet r c n and t : P( t) St at st cs 7 t t P t e wth t e d no counts n t t P( ) t e d e ln[ P( )] ln[ CL] t t t nor alzed P nor alzed P Upper lt The hgher t he conf dence level CL ( CL ), t he lar ger t he upper lt f or a gven t e t nspect ed. Reduce lt by easur ng f or longer per od.

18 Mau Lkelhood Measur eent of cor r elat ons bet ween obser vables y and : {, y =-} Hypothess: y() =f(c,,c ; ). Paraeters defnng f: {c,,c } n dof =- degr ees of f r eedo f or a f t of t he dat a wt h f. P( c,.., c ; ) ep y f ( c,.., c ; ) for every data pont St at st cs 8 c Maze sult aneous pr obablt y P( c,.., c ) P( c,.., c ; ) j y y f ( c,.., c; ),.., c : Mnze ch-squared by varyng {c,,c }: / c = j ep y When s as good as can be?

19 Mnzng St at st cs 9 Eaple: lnear ft f(a,b;) = a + b to data set {, y, } Mnze: a, b a a a a, b b a, b : b y b y y a b y a b y a b a y a b Equvalent t o solvng syst e of lnear equat ons y ad d c ad a b bd c d d D d d c d d c b D c d D d c a b D D

20 Dst rbut on of Ch- Squareds Dstrbuton of possble f or dat a set s dst r but ed nor ally about a t heor et cal epect at on (f unct on) wt h n dof degrees of freedo: St at st cs P( u ) P( u ) P( u 3) P( u 4) P( u 5) P(, n ) P ( ) d dof Ch-Squared Dstrbuton < > ndof = u u:= n dof Should be P.5 for a reasonable ft n dof e Pn ( ) d d dof n dof n dof dof dof n n n ( n) n! St r lng s forula n n dof.57 e n (.833 / n) Reduced : r dof For r n ( ).5 Confdence 5%

21 CL f or - Dst rbut ons St at st cs

22 Correlat ons n Dat a Set s Cor r elat ons wt hn dat a set. Eaple: y sall whenever sall St at st cs y uncorrel. P(,y) Punc (, y) P( ) P( y) y y ep 4 y y correlated P(,y) P corr cot (, y) ep y y y y y y y y y y 4 y y y y y P(, y) ddy covarance y r correlat on coeffcent r y y y y

23 Mult varat e Correlat ons c j ntal guess surface Dfferent search strateges software packages LI FI T, MIUIT,. St at st cs 3 search path c

24 Correlat ons n Dat a Set s uncer t ant es of deduced ost lkely par aet er s c (e.g., a, b for lnear f t ) depend on dept h/ shallowness and shape of t he surface St at st cs 4 c j uncorrelated surface c j correlated surface c c P(, y) P( ) P( y) r r y y : y y r y y y : y y ; ep j y y covarance n n y j y y 4 y