3: Central Limit Theorem, Systematic Errors

Transcription

1 3: Central Lmt Theorem, Systematc Errors 1 Errors 1.1 Central Lmt Theorem Ths theorem s of prme mportance when measurng physcal quanttes because usually the mperfectons n the measurements are due to several causes, nstead of only one. It deals wth the behavor of a varable whch s tself the sum of several other varables. The followng statement s not rgorous, but has a clear physcal meanng: Central lmt theorem: Let x, = 1,..., n be n ndependent varables, each of them dstrbuted accordng to any law of mean µ and varance σ. The expectaton for X = x s µ, the varance s V (X) = σ and the dstrbuton of X becomes Gaussan as n grows, provded the σ are of the same order of magntude. Proof: The proof of the frst two statements s very easy. The expectaton of x s µ, no matter the x s are ndependent or not. The varance of the sum s, takng nto account the ndependence of the varables: V ( x ) = < ( x < x > ) > = < ( x µ ) > = < [ (x µ )] > = < (x µ ) > + < (x µ )(x j µ j ) > j = < (x µ ) > + < (x µ )(x j µ j ) > j = σ + cov(x, x j ) j = σ The proof of the thrd statement s much more subtle: let s recall that we ntroduced the characterstc functon of a dstrbuton as Φ(t) =< e tx >. Around t = 0, ths expresson can be expanded as: Φ(t) = 1 + t < x > + (t)! Takng the logarthm of ths expresson yelds: < x > (t)n n! < x n > +... log[φ(t)] = (t) k 1 + (t) k (t)n k n +...! n! where the k s are called the cumulants of the dstrbuton. For example, k 1 = < x > k = < x > < x > k 3 = < x 3 > 3 < x > < x > + < x 3 > 1

2 From the defntons of the cumulants, one sees that: k 1 = µ, k = σ, k 3 = µ 3, but k 4 µ 4 If x x = x + a, then Φ (t) = Φ(t)e ta and log Φ (t) = log Φ(t) + (t)a. Thus k 1 = k 1 + a and k = k, for > 1. If x x = bx, then snce each k s a sum of terms n < x k > < x j >, wth = j + k, k = b k. The cumulants of the convoluton product of two dstrbutons (whch s the dstrbuton of the sum of the two varables) are the sums of the cumulants of the two orgnal dstrbutons. We already saw that the characterstc functon of the Gaussan dstrbuton s thus Φ(t) = e tµ σ t log Φ(t) = (t)µ + (t) σ! whch means that k 1 = µ, k = σ, and k = 0, for any >. In order to prove the Central Lmt Theorem, one has to consder the cumulants of the dstrbuton of X = x µ nσ where σ s the average standard devaton. The reason for ths normalsaton s that one s comparng the dstrbuton of x wth the normal standard dstrbuton. The cumulants of x are K r = k r = nk r, where k r s the average r th cumulant, and the cumulants of X are nk r (nσ ) r = nk r (nσ) r/ For r >, ths goes to zero as n grows. How fast the convergence s, depends on the orgnal cumulants k r,.e. on how close the orgnal dstrbutons are to the normal dstrbuton. Ths can be llustrated by the followng: Wrte a program to see how fast n the conver- Exercse: Central Lmt Theorem at work gence of the CLT s, for the followng dstrbutons: The exponental dstrbuton The unform dstrbuton Half of the varables whch are summed come from an exponental dstrbuton, and half of them from a unform one. The soluton can be found here: --> CLT.m

3 1. Weghted measurements Assume than we run an experment measurng N tmes the same quantty x of true value µ, each tme wth the same uncertanty σ. Because of the central lmt theorem, the average x = ( x )/N has an expectaton value of µ and a varance of V (x) = 1 σ N = σ N Ths s true whatever the dstrbuton of the measurement error s. Moreover, the dstrbuton of the average x wll tend to be Gaussan. If the orgnal dstrbuton s Gaussan, the average wll always be exactly dstrbuted accordng to a normal law. Weghtng measurements s the procedure one has to use when the varous measurements x possess dfferent uncertantes σ. To combne the values t seems natural to weght the measurements such as better ones are gven more weght than the poorer. The proof of the method wll be gven later on but t s smple to show ntutvely that the correct weghtng procedure s to form the quantty: x = x /σ 1/σ Let s assume than the measurements were all orgnally performed wth the same uncertanty σ, then grouped nto groups, each group havng ts own estmated average x and uncertanty σ. Now assume that one of the grouped measurements s, say, twce better than another one j: σ = σ j /. Ths means that group conssts of 4 tmes more elementary measurement j. Therefore, f one would have measured x as ( x )/N, group would have contrbuted 4 tmes more than group j. Ths can also be seen n the smple example of averagng two measurements: let x 1 be the frst one, v(x 1 ) = σ1, and x, v(x ) = σ the second one. One s tryng to mnmze the varance of w = αx 1 + (1 α)x, when α vares. v(w) = α σ 1 + (1 α) σ reaches ts mnmum for dv(w) dα = ασ 1 (1 α)σ = 0 whch means that the weghts α and 1 α are nversely proportonal to the varance of the measurements: Exercse: averagng counts α = σ σ 1 + σ A long lvng radoactve source gves 389 counts n the frst mnute and 43 n the second mnute. What s the best combned result? The strength of a long-lvng partcular radoactve source was measured three tmes wth the followng results:.31 ± 0.11,.56 ± 0.15,.4 ± 0.07µC Can one answer the queston: what s the best value and uncertanty for ts strength? 3

4 Exercse: smulatng the weghtng procedure Wrte a MATLAB program to smulate the followng two stuatons: measurng 10 tmes the same quantty X of true value X =, each tme wth a measurement error ether normal wth σ randomly chosen between 1 and 5, or unform between b/ and +b/, b beng randomly chosen between 1 and 5. In each stuaton, generate artfcal data and estmate X n two ways: through a nave averagng and through a weghted average. Repeat the exercse to smulate 100 experments, plot the hstograms of the estmated X and compare the varance. Is a lot ganed by the weghtng method? The soluton can be found here: --> WEIGHT.m 1.3 Systematc errors Systematc errors are errors whch affect the same ways repettve measurements of a gven quantty. They have two nasty propertes: they prevent mprovng the precson by repeatng the measurements, and they produce effects from the non-ndependence of measurements at dfferent ponts. Beng nternally consstent, unlke random errors and statstcal uncertantes, they are often qute dffcult to spot. But, n spte of popular belefs, once you know what they are, you can handle them wth standard statstcal methods. As an example, let us dscuss the case of two varables x 1 and x havng the same systematc error S. We can model the stuaton as f x 1 and x conssted of two parts, x 1 = x 1R + x S and x = x R + x S, wth x 1R and x R ndependent of each other, wth varance σ1 and σ, respectvely, whle x S has a varance of S. One also assumes that x R and x S are ndependent. The varance of x 1 can be expressed as: V (x 1 ) = < x 1 > < x 1 > = < (x 1R + x S ) > < x 1R + x S > = < x 1R > + < x S > + < x 1R x S > < x 1R > < x S > < x 1R > < x S > = σ 1 + S The covarance between x 1 and x s gven by: cov(x 1, x ) = < x 1 x > < x 1 >< x > = < (x 1R + x S ) < (x R + x S ) > < x 1R + x S >< x R + x S > = < x 1R x R > + < x 1R x S > + < x S x R > + < x S x S > < x 1R >< x R > < x 1R >< x S > < x S >< x R > < x S >< x S > = cov(x S, x S ) = S The net result s that the covarance matrx can be wrtten: ( σ V = 1 + S S S σ + S ) Note that f nstead of a constant systematc error, one deals wth e.g. systematc errors proportonal to the measurement (or the true value, but for small errors the dfference s neglgble): S = ɛx, the dstrbuton for x 1S and x S are no longer equal but absolutely correlated (ρ = 1), and the above analyss gves ( ) σ V = 1 + ɛ x 1 ɛ x 1 x ɛ x 1 x σ + ɛ x 4

5 A functon of two quanttes wth common systematc error We now address the followng queston: two quanttes x 1 and x have been measured wth a common systematc error. What are the systematc and statstcal errors of any functon f(x 1, x )? We saw n..1 that the law of propagaton of errors, s ths case, s: V (f) = =1 j=1 V j x x j = (σ1 + S )( ) + (σ + S )( ) + S x 1 x x 1 x = [ σ 1 ] + [ σ ] + [ S ( + ) ] x 1 x x 1 x Exercse: Product and rato of quanttes wth common systematc error Assume that the quanttes x 1 and x are measured wth the same systematc error, but wth dfferent statstcal uncertanty: x 1 =.05 ± 0.16 ± 0.11 x = 1.06 ± 0.1 ± 0.11 What are the systematc and statstcal uncertantes of x 1 + x, x 1 x, x 1 x, and x 1 /x? 5