Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede Some mportant probablty dstrbutons: Unform Bnomal Posson Gaussan/ormal The Unform dstrbuton s often called U( a, b ), hch stands for unform dstrbuton on the nterval a < b. For U (,) the P.D.F. s: f < f elsehere f d f d Then: [ ] ˆ [ ] E f d E f d ( ˆ) ( ˆ)( ˆ) f ( ) var σ E[ ] E[ ] E[ ] E[ ] 4 Here e have plotted computer-generated values of U (,) hch has a mean of zero and standard devaton of. By tself, the Unform dstrbuton s not so mportant (although t s smple), but t can be used to generate a sgnfcant number of other probablty dstrbutons and t s also easy to smulate on a computer. P598AEM ecture otes 7
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede A frequently-used trck n computer programmng {hch actually s not a trck has some deep physcs } s to sum several random numbers r, hch have been chosen from U (,). For eample, consder the random varable dstrbuton at X X 6 (the 6 centers the mean of the ). Many such values of X are plotted n the fgure belo, along th an overlay of the functon / G e, the Gaussan/normal dstrbuton th ˆ and π σ (n.b. sometmes, ). Apart from the tals, the values of X tend G s called to follo (,) and ths s a poor man s method of generatng random numbers dstrbuted accordng to (,). entres of X 6 here s a random number dstrbuted n accordance th U(,). The Gaussan/normal dstrbuton shon has mean zero and standard devaton. X P598AEM ecture otes 7
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede et S. Then: Sˆ ˆ for random numbers chosen from (,) U. If the numbers are truly random (more on ths later ) they ll be ndependent, and thus: σ S σ S S Thus, the varable X ll have ˆ X and σ σ X S,.e. t ll be normalzed. ater on e ll prove that: lm (,), the Gaussan/normal dstrbuton. In order to understand ths behavor, e begn by calculatng the P.D.F. assocated th the sum of pars of ndependent random varables u +. (ote: e treated the case of the sum of to Gaussan/normally-dstrbuted varables n a prevous lecture. o e do the general case.) In addton to u +, e also set v here: g( u, v) (, ) f uv J and here: So g( u, v) f (, ) f ( ) f ( ). Then set (, ) (, ) g u v du dv f d d. uv J (because The P.D.F. of u s then the margnal dstrbuton: u v u v and are ndependent)., thus here: J. or: ( + ) ( ) Gu g uv, dv f f dv Apply ths to the stuaton here and Gu f v f u v dv are ndependent random varables obtaned from (,) U : f < f f < f.e. (for both and ) Then: ( + ) ( ) ( ) ( ) G u f v f u v dv f v f u v dv f u v dv Ths last ntegral on the RHS of the above equaton may look trval, but n fact t s not... P598AEM ecture otes 7
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede We consder 4 cases: a.) u < : G( u ) snce f ( u v) b.) u < :. u u G u f u v dv dv+ dv u u c.) u < : G u f u v dv dv+ dv v u u d.) u : G( u ) snce u u f u v. f () G(u) u Sum S of pars of random varables dstrbuted as U(,) plotted as X S. A straght lne has been ftted to X >. et, e add, hch s also dran from U (,) to the above ( + ) ( ) ( + + ) ( ) G u : H u G z f z dz here: + + and: z u + H G z f z dz thus: u+ z+ z+ y,.e. y Hence: z y or: y z and thus: dy dz. Then: ( ) ( ) ( ) H G y f y dy G y f y dy Thus, e see that H( ) for < or >. X P598AEM ecture otes 7 4
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede < f H( ) + 6 f < ( ) f < A bt of ork shos that H()..5 The dstrbuton s startng to look normal : S entres of X here S s the sum of a trplet of random numbers dstrbuted as U(,). The curve s a parabola ftted to the regon < <. Suppose e add many U X, ' s? What happens for S X n the lmt? It (rapdly) gets too messy/too tedous to contnue usng ths ntegral method... P598AEM ecture otes 7 5
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede To help us n the study of probablty dstrbutons, e ntroduce a ne tool knon as the k k E e, here k s the Fourer conjugate varable to. Characterstc Functon [ ] If the P.D.F. of the random varable s f ( ), ts Fourer transform s the Characterstc Functon: [ k k ] k E e e f d The nverse Fourer transform of the Characterstc Functon (assumng t ests) s the P.D.F.: For the case of dscrete random varables th dscrete nverse Fourer transform: The characterstc functon ( k ) k f e π th assocated probabltes k k E[ e ] e P( ) k ( ) P e k P : s mportant/useful n combnng dstrbutons and recognzng partcular dstrbutons. We begn to eplore some of the features of the characterstc functon: et us nvestgate a ne random varable a + b : ( + ) ( + ) k E[ e k a b ] e k a b f d e kb ak a+ b In partcular a, b ˆ : k ˆ k ˆ kˆ ˆ k E[ e ] e f d e k Characterstc Functons are usually smpler to ork th than Probablty Densty Functons. If and P.D.F. s are ndependent, such that the P.D.F. f (, ) f f f and f ( ), then: () and e kno the k( + ) ( ) [ ] k +, k k + t E e d d e f de f d e f for ndependent random varables and. So e see that: + P598AEM ecture otes 7 6
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede Compare ths smple result for the characterstc functon to the result that e obtaned (above) for the P.D.F. of the random varable u +. As a second varable, e agan let v. Then: g ( u, v) du dv f (, ) d d f ( ) d f ( ) d here: g( u, v) and here (agan): uv J u v u v So: g( u, v) f (, ) f ( ) f ( ) (because The P.D.F. of u s then the margnal dstrbuton: (, ) G u g u v dv f f dv or: ( + ) ( ) G u f v f u v dv (, ) f uv J uv, thus here: J. and are ndependent). As e have already seen earler n ths lecture, ths functon s sgnfcantly more complcated k k k. than + For the Unform dstrbuton U (,) : k k k k k e e E[ e ] e f d e d k k k e + k Then: Generalzng to,,, ndependent random varables: ( k ) Hoever, hat e really care about (n the lmt ) s: k e k X S a b P598AEM ecture otes 7 7
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede kb From e ( ak) o: But: e obtan: a+ b {} k k e k k e k Here, the subscrpt {} stands for { } k e ln k + ln k k + k + k + e! + k + k +...!! k k k + ε f Then usng the Taylor seres epanson: ln( + ε) ε ε + ε e obtan: ln k Thus: ln + k + k +!! k! k k 6 4 k + k k k k O + lm { ln } k and hence: lm k e Thus, e see that the characterstc functon ( k ) a b assocated th X approaches k e as. P598AEM ecture otes 7 8
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede Fnally, as the P.D.F. of X s ndeed the Gaussan/normal dstrbuton (,) : kx kx k X e k e e e G X π π π Ths result s one eample of the Central mt Theorem. In ths case, e see that f the ndependent random varables are unformly dstrbuted n the nterval (,), ther (normalzed) sum the ormal (Gaussan) dstrbuton (,) as. We net prove a eaker form of the Central mt Theorem: et the,,,,..., be a set of ndependent random varables th the same P.D.F. f ( ) and assume that the values of ˆ and σ are gven/specfed. Then, n the lmt the sample mean follos a normal dstrbuton th epectaton value ˆ and varance σ. (ote here that the P.D.F. f ( ) s not specfed.) We begn by replacng by ˆ random varables. Then: as the, and then treatng the ˆ (,,,..., ) k ( k )! k E[ e ] + k E[ ] + k E[ ] + k σ + O If e no set o look at: u ˆ, then: σ σ k kb usng e ( ak) u a+ b σ u σ k k + O ˆ. Ths has a characterstc functon of the form: k k u + O Then: k k ln ln + O P598AEM ecture otes 7 9
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede Agan, usng the Taylor seres epanson: ln( + ε) ε ε + ε e see that as : { } lm ln k k and so (agan) e see that: ˆ lm has P.D.F. σ f( ) e.e. s (,) π dstrbuted. o e need to rerte ths n terms of the mean : ˆ ˆ σ σ But the sample mean. σ So: ( ˆ ) ˆ σ To change varables from to e rte: f ( d ) g( d ) and: g ( σ ) e f( ) f( ) π J( ) d d σ π σ ˆ ( ˆ ) σ ( ) ( ) e ( ) In other ords, the sample mean of a large number of ndependent random varables th the same P.D.F. and gven ˆ and σ follos a Gaussan/normal dstrbuton th epectaton value ˆ and varance (of the mean) σ σ. The specfc form of the P.D.F. of does not matter! Important note: ths result s rgorously true only n the lmt of. Ths statement can be made even eaker: The mean of a large number of ndependent random varables follos a Gaussan/normal dstrbuton even hen all of the do not have the same P.D.F. (We ll not prove ths here.) Fnally, e state (but agan do not prove) the Central mt Theorem: Suppose e are gven a collecton of ndependent random varables (,,,..., ) each follong a P.D.F. f ( ) for hch ˆ and σ are defned. (The f ( ), ˆ and the same or dfferent.) Then, σ may be ll be normally/gaussan dstrbuted as, th some restrctons. In partcular, no σ may be very much bgger than all of the rest. To be precse: lm ˆ σ (,) P598AEM ecture otes 7
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede Calculaton of the Moments of a P.D.F usng the Characterstc Functon: In P598AEM ect. otes e dscussed computng the moments e.g. of a contnuous P.D.F. f ( ) : The th algebrac moment s defned as: α of a contnuously-dstrbuted random varable taken about [ α E ] f d The th central moment μ of a contnuously-dstrbuted random varable taken about the mean ˆ α s defned as: μ E[( ˆ) ] ˆ f d The Characterstc Functon assocated th the P.D.F. f ( ) s: and: [ k k ] k E e e f d k ˆ k ˆ kˆ ˆ k E[ e ] e f d e k Then the th algebrac moment α of the contnuously-dstrbuted random varable taken about k : can also be defned n terms of (dervatves) of the Characterstc Functon α E [ ] f d () d k k d e f d + k k k () () e f d f d The th central moment μ of a contnuously-dstrbuted random varable taken about the true mean ˆ α can also be defned n terms of (dervatves) of the Characterstc kˆ k e k : Functon μ ˆ kˆ ( ) d e k d ˆ ˆ ( ˆ E[ ] ) f d () k k k( ˆ ) d e f d k( ˆ ) () + () () ( ˆ ) k k ( ˆ) e f d f d P598AEM ecture otes 7
Fall Analyss of Epermental Measurements B. Esensten/rev. S. Errede Then the frst fe moments assocated th the P.D.F. f ( ) are: True mean: ˆ α E[ ] E[ ] f d Varance: d k ˆ σ μ α α E ˆ ( ˆ) f d var [ ] α ˆ Skeness γ : μ μ μ μ γ μ var σ σ ( ) ( ) ˆ ˆ + E[( ) ] ( ˆ) f d μ α αα α α ˆα + ˆ Ecess Kurtoss γ : μ4 μ4 μ4 μ4 γ 4 4 4 4 μ var σ σ ( ) ( ) d d 4 4 4 4 4 ˆ 4 4 4 + 6 ˆ E[( ) ] ( ˆ) f d 4 μ α αα α α α α 4ˆα + 6ˆ α ˆ 4 4 d k k k P598AEM ecture otes 7