Ex x xf xdx. Ex+ a = x+ a f x dx= xf x dx+ a f xdx= xˆ. E H x H x H x f x dx ˆ ( ) ( ) ( ) μ is actually the first moment of the random ( )

Transcription

1 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede The Epectation Value o a Random Variable: The epectation value E[ ] o a random variable is the mean value o, ie ˆ (aa μ ) For discrete i, [ ] E is the sum o all i, weighted by their associated probabilities Discrete i : E [ ] ˆ μ P N i= i i P : For continuous, E [ ] is the integral over all, weighted by the probability density unction o, ( ): Continuous : [ ] ˆ μ E d Note: ˆ is not a random variable it is a single, well-deined number that characterizes the true mean o the particular/speciic distribution o a random variable It is clear that or any constant, a: N N N E+ a = + a P = P + a P = ˆ + a Discrete: [ ] i i i i i i= i= i= = E+ a = + a d= d+ a d= ˆ + a Continuous: [ ] In act, any unction o, H( ) has an epectation value: Discrete: EH [ ] Hˆ H P N i= Continuous: [ ] ˆ E H H H d The true mean/epectation value E [ ] ˆ μ is actually the irst moment o the random variable s probability distribution, relative to/taen about the origin = 0 The L th moment o the random variable taen about an arbitrary point = c is deined as: i i = i μ L E[( c) L ] nb Moments taen about = 0 are nown as algebraic moments Moments taen about the true mean = μ are nown as central moments In particular, the nd moment o taen about/with respect to the true mean ˆ is: μ [ ] var The quantity ˆ E the variance o μ var = is nown as the dispersion, or standard deviation o P598AEM Lecture Note 03

2 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede The true mean ˆ tells us where (in ) the distribution is important ie ˆ tells us where the random variable is most liely to be ound) The dispersion/standard deviation is a measure o the spread (or width) o the distribution over the space o small variance large variance The two distributions in the above igure have the same epectation value/true mean E [ ] = ˆ = μ, but obviously have very dierent widths and hence have dierent variances Eample o the Uniorm Distribution U(0,) in the Range = 0 to = : The Uniorm Probability Density Function (PDF) U ( 0,) is: ( ) i 0 < = 0 otherwise This is correctly normalized, ie (ie the area under the curve o d = d = d = The epectation value/true mean o U ( 0,) is: ˆ vs is = ) = d= d= The variance associated with U ( 0,) is: var = = ˆ = ( ) d d 0 0 We can certainly calculate the above variance or U(0,) directly, however, in general it is requently oten easier to calculate, i the variance is irst transormed in the ollowing way: var = = ( ˆ) d ( ˆ ˆ ) = + d ˆ ˆ ˆ ˆ = + d d d = + d d d [ ] ˆ [ ] ˆ [ ] ˆ = E E+ = E + or the U ( 0,) distribution: var = = E [ ] E [ ] = ˆ = E [ ] ˆ P598AEM Lecture Note 03

3 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede So or the U ( 0,) distribution: U(0,) var d = = = = 0 = ˆ = This means that i we perorm a measurement o a quantity that results in the nowledge that it can be anywhere (with equal lielihood/equal/lat probability) within the range (0,), our best estimate o its location is that is at the true mean, = ˆ = 05 with a standard deviation o = 09 More generally, or the lat/uniorm distribution U 0, L = L: L L mean: ˆ = variance: = standard deviation: = I the allowed range o is a b, it is also easy to show that U( a, b) ( b a) L =, and that: b+ a b a b a mean: ˆ = variance: = standard deviation: = Eample The HEP Hodoscope: In particle physics eperiments, a planar array o scintillation counters ( hodoscope ) is oten used to detect the passage o ionizing (charged) particles I a particular counter is hit (and that is all that on one nows), then this inormation can be considered a measurement o (say) the coordinate o the trajectory at the plane o the hodoscope The result o the measurement is that ˆ is the -coordinate o the center o the particular counter that was hit by the particle The standard deviation o the measurement is = L 09L, where L is the lateral width (in the -direction) o the counter, as shown in the igure below: L P598AEM Lecture Note 03 3

4 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede This eample is also relevant or determining the uncertainty in ADC count data in the regime o noise-ree data eg a -bit ADC measures quasi-dc voltages over a ±50 volt range bits = 4096 Note that: 0 Volts/4096 bits = 44 mv/adc count For noise-ree {or nearly noise-ree} ADC data, in the absence o using so-called ADC dithering techniques (adding and systematically {or randomly} varying a small voltage oset), then ADC V = 44mV = 070mV = ADC counts, corresponding to Eample The Cauchy Distribution: = π L + L = π L + L /πl 8L 0 +8L A careul chec shows that: d= + OK + L + ˆ = d= ln( + L ) Undeined!!! π but: Obviously, the epectation value o is/should be ˆ = 0 Hmmm There are mathematical complications with this PDF!!! Using ˆ = 0, we can calculate the variance o the Cauchy Distribution (using a table o integrals): var = = L d= π L ininite More mathematical problems! d L + inite Ininite!!! P598AEM Lecture Note 03 4

5 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede In order to avoid these diiculties, in practice a truncated Cauchy distribution is used: L c i < X = L + 0 otherwise X X d requires: c = The requirement that: = + L tan ( X / L) + X Now: d ( L ) (Note that or X, c Lπ as we epect) ˆ = = L + X ln 0 X π + = X and var = is inite as well More About the Moments o a Probability Distribution Function / PDF The st central moment (ie taen about the true mean ˆ μ ˆ ˆ ˆ ˆ E [ ] = E [ ] E [ ] = = 0 Higher moments o a probability distribution are sometimes useul = μ ) is not interesting, since: For eample the coeicients o sewness and urtosis, (Gr urtos = bulging/swelling ) respectively are the 3 rd and 4 th standardized central moments o a distribution (ie also taen about the true mean ˆ = μ ): Sewness: Ecess Kurtosis: γ μ E[( ) ] E[( ) ] = = = μ ˆ { E[( ) ]} = ˆ 4 4 μ4 E[( ˆ) ] E[( ˆ) ] γ = 3= 3= μ ˆ { E[( ) ]} = ˆ The 3 in the ecess urtosis deinition is added so that γ = 0 or a Gaussian (aa normal) 4 probability distribution Without the 3, regular/normal urtosis is deined as μ4 μ P598AEM Lecture Note 03 5

6 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede γ tells us whether the PDF Sewness γ < 0: sewed toward < ˆ γ = 0: symmetric about ˆ γ > 0: sewed toward > ˆ is: γ < 0 γ = 0 γ > 0 ˆ ˆ ˆ 3 Obviously, the long tail dominates in E[( ˆ) ] A probability distribution (eg the Gaussian/normal distribution see below ) that has γ = is called mesourtic, or mesourotic (Gr mesos = middle ) zero urtosis 0 Positive urtosis (γ > 0) indicates a peaed/slender/narrow probability distribution near the mean with abnormally long tails, and is called leptourtic, or leptourotic (Gr leptos = slender/thin ) Negative urtosis (γ < 0) indicates a lattened/wide/broad probability distribution near the mean with abnormally shortened tails, and is called platyurtic, or platyurotic (Gr platys = broad/lat ) Stoc maret investors are very interested in these particular higher moments (sewness & urtosis) o various probability distribution unctions Important Note: Each/every measurement o a random variable contains inormation about the PDF ( ) rom which it originates and its epectation value/true mean ˆ, variance var PDF s higher order moments =, and the In the limit o an ininite # o measurements N o the random variable, the PDF and all o its associated moments are precisely nown P598AEM Lecture Note 03 6

7 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede Any random variable can be normalized or standardized and can thus be reduced to a dimensionless quantity Suppose that the random variable has PDF ( ) and has a well-deined epectation value / true mean ˆ and variance Consider a new random variable deined as: u ( ˆ ) Aside: in general, i c is a constant, then: Ecg ceg nb dimensionless quantity [ ] = [ ] ˆ E[ u ] = u = u d= d= ( ) d= 0, ie u ˆ = 0 Thus: ˆ ˆ and: ( ˆ) var u = u u d= u d uˆ u d + uˆ = ( ˆ) = = d var = = u = d u d Thus, the random variable ( ˆ) u has: Epectation value/true mean Eu uˆ Variance [ ] = = 0 var u = u =, hence standard deviation = u P598AEM Lecture Note 03 7

8 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede The -D Gaussian (aa Normal) Distribution: One o the most common PDF s that is encountered in the everyday world o doing eperimental physics is the Gaussian (aa normal) probability distribution unction (PDF): u Gu = e with: = π G(u) G(0) = / π G(±) = 06G(0) 0 + u It is easily veriied that i u has the above PDF, then the epectation value/true mean o the Gaussian/normal distribution is E[u] = 0 (by symmetry) and its standard deviation, u = In terms o probability, the probability that u will be ound in an interval du is: du at u = 0 π P = e du at u =± π The ratio G(±)/G(0) is e = Later we shall show why the normal distribution is so universal in eperimental physics In many situations in eperimental physics, we need to describe a normally distributed random variable with epectation value / true mean ˆ and standard deviation We can also invert the transormation u = ˆ : d = G u du = e π The PDF o satisies: ( ˆ ) / d ( ˆ ) / = e π ie Shited Gaussian distribution, centered at = ˆ with standard deviation P598AEM Lecture Note 03 8

9 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede ( ˆ) = = π ( = ˆ ± ) ( ˆ ) = 06 = The N-D Probability Distributions Functions, N-D Cumulative Distribution Functions: In physics eperiments, we oten measure more than one property o a system (eg the three spatial coordinates o a particle s position, its 3-momentum (or velocity), the energy and/or lietime o an ecited atomic state, etc) An eperiment may then result in dierent random variables, which can be conveniently plotted on mutually orthogonal aes (ie a mathematical hyper-space consisting o orthogonal dimensions) and treated as i they were the components o a -dimensional vector First, let us consider only two such random variables and y The -D Cumulative Distribution Function: F ( XY, ) Prob ( < Xand y< Y) y The -D Probability Density Function: ( y, ) F( y, ) I the random variables and y are independent, then ( y, ) is separable in & y, ie ( y) = ( y) & y have no correlations (ie are uncorrelated), y Then: P ( a b b d, c y d ) d dy (, y ) + + < < = with the requirement: d dy ( y) a ˆ ˆ ˆ + c where the, y integrals run over the allowed values o and y: a b + and: c y d +, = P598AEM Lecture Note 03 9

10 Fall 03 Analysis o Eperimental Measurements B Eisenstein/rev S Errede In certain eperimental situations, we may not care (or even now) what y is, + hence we integrate y over its entire range and get: g = ( ydy, ) b In this situation: (, ) P a < b y< + = g d Thus we see that g( ) is also a PDF; g( ) is nown as the marginal distribution o Change o Random Variables: Suppose we have random variables (,,, ),,, a, with corresponding PDF We may wish (or need) to mae a change variables rom ( ) to where each o the y = y (,,, ) i i,,, y, y,, y, For the case o a single random variable, or a change o variables y = H, rom P598AEM Lecture Notes, p 8, the relation g ( y) dy = d leads to the relation (here): g y = where: H H = dy d = y-slope nb absolute value For random variables, this generalizes to: g y y y (,,, ) = J (,,, ) nb absolute value where J is the Jacobian: yy y J y y y y y y y y y nb The Jacobian J is the determinant o the matri o derivatives y i j ie compute the individual entries o the matri o partial derivatives y i j then compute the determinant o this matri, then tae the absolute value o the determinant, P598AEM Lecture Note 03 0